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Bulletin 745 (technical)
April 1971
The Dairy Cow Replacement Problem
An Application of Dynamic Programming 
Blair J. Smith
Agricultural Experiment Stations
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
J. W. Sites, Dean for Research
CONTENTS
Page
INTRODUCTION ...................................... ............... 3
REVIEW OF RELATED REPLACEMENT STUDIES ..~.............. ... 4
DEVELOPMENT OF REPLACEMENT VARIABLES
AND PARAMETERS .........~..... .......................... 6
Determinants of Milk Production ................. ................ 6
Size of Cow ...... ...... .. ... ......... ..... ..... .......... 6
Age at First Calving ............................ .............. .... 7
Age at Calving Each Particular Lactation ........ ............ 7
Length of Dry Period ......................... ...... ........ 8
Length of Calving Interval ................................... 8
Milk Yields of Close Relatives and HerdMates .......... .......... 9
Production by the Same Cow in Prior Lactations ................. .. 9
Miscellaneous and Minor Factors ............ ............ 11
Empirical Quantification and Selection of Predicting Variables ... 11
Basic Cow Data .............................. ...... ... . ............ .... 11
Screening and Choosing Available Variables .................. ..... 12
Analysis of Previous Calving Interval and Prior Production ..... 12
Generalized Production Prediction Model ............ .............. .... 16
Reasons for Cow Removals and Development of Transitional
Probabilities .......................... ........ ... .... 17
Characteristics of the Replacement Cow ............. ... .. 20
Economic Components of the Replacement Problem .............. ... 22
Quantification of Variable Costs and Returns ........... .. 22
Value of Milk Produced ............... ... ... ..... ... ..... . 22
Value of Calves Born .......................... ......... 23
Salvage Value of the Cow ............................. 24
Depreciation in Value of Cow .......... ...... ............ 24
Interest on Capital Value of Cow .................. ...... 24
Feed Costs ..... .... ..... ........ ...... ...  .. 24
Net Variable Returns  ....... . 28
DAIRY COW REPLACEMENT MODEL SPECIFICATIONS ......... 29
General Specifications ... ........... ........ ....29
The Dynamic Programming Replacement Model ....... .... .. 32
Method of Incorporating Production in Prior Lactations ..... 37
EMPIRICAL ANALYSES OF THE REPLACEMENT MODEL ....... 38
General Com m ents .. ..... .. .... ............ .... ...... ........ ............. ........... 38
Summary and Comparison of the Results of the Net Returns Runs 41
Results of the Production Run with Comparisons to the Net
Returns Runs ....... ...........~............... ........ .... ..... ... 42
Operational Employment of the Model and Costs of Computation ... 43
SUMMARY .................................... .......... 44
LITERATURE CITED .........~.. ......... ........... 47
ACKNOWLEDGMENTS ....... ......... .. . ... .. 50
THE DAIRY COW REPLACEMENT PROBLEM
An Application of Dynamic Programming
by
Blair J. Smith'
INTRODUCTION
Dairy husbandmen have long subscribed to the trichotomy
of emphases in dairy herd management which are suggested by
the old maxim "breed, feed, and weed." The degree of formal,
systematized attention which has been devoted to "weed" de
cisions has generally lagged that given to the other two, how
ever. This research was undertaken in the hope that some re
dress of the imbalance of the emphases might be accomplished.
A survey of the literature on the culling of dairy cows re
vealed that, although a great deal of importance has been at
tached to past production records, there is a notable lack of
operational evaluative processes which both take account of
other cow characteristics and are projective in nature. There
have recently been a few projective studies which do consider
factors other than past production, but none of these can qualify
as being operationally employable in their present form.
There are two distinct phases in the development of dairy
cow replacement models. The first is the identification and quan
tification of the variables and relationships which "explain"
differences in the expected performance of cows. The second
phase is the development of a model which makes use of the
relevant relationships to identify an optimal2 ReplaceKeep de
cision for any cow and potential replacement with any combi
nation of observed characteristics.
The basic data used in phase I of the present study were
from lifetime histories of 369 Jersey cows which had been in
the University of Florida Experimental herd, supplemented by
information from other sources. Efforts related to phase II
of the research upon which this report is based culminated in
the development and evaluation of a dynamic programming re
'Assistant Economist in the Department of Agricultural Economics at
the University of Florida at the time of this study, now Associate Professor
in the Department of Agricultural Economics at the Georgia Station of the
University of Georgia College of Agriculture Experiment Stations, Experi
ment, Georgia.
SThe definition of an "optimal replacement policy" is always within the
context of the decision model within which it is determined. Other formula
tions of the decision model or other specifications in a given model would
probably yield different optimal replacement policies.
placement decision model. The maximizations of the present
value of both expected production and expected net returns were
used as objective criteria in the model evaluation. Though little
difference in the optimal replacement policies for each criterion
was found, the use of the net returns criterion is the more de
fensible in an economic context.
REVIEW OF RELATED REPLACEMENT STUDIES
Replacement models are decision models inasmuch as the de
cisionmaker may choose among two or more alternative courses
of action to affect the output or condition of the system being
modeled. Work by Bellman (1957) made it possible to cast the
replacement problem in a sequential, or multistage, decision
making framework, and was a milestone in the development of
replacement theory. Howard (1960) incorporated stochastic
(Markov) processes into the dynamic programming model of
Bellman. It is from the work of these two men that most recent
related studies of replacement draw their theoretical and meth
odological concepts and insights.
The first known formal application of a dynamic program
ming approach to the replacement of a biological asset in an
agricultural setting was carried out by White (1959) and re
ported more fully by Halter and White (1962). Their appli
cation was to a caged layer enterprise. Burt and Allison (1963)
considered the problem of the optimal sequence of plant wheat
or let fallow in a dryland farming area of the Great Plains.
The plantfallow decision in their problem is analogous to the
keepreplace decision in replacement problems. For a more
generalized orientation and overview of the replacement prob
lem and appropriate analytical techniques the articles by Faris
(1960) and Burt (1965) are useful.
Two major studies of dairy cow replacement have been made
prior to the one reported here. Jenkins and Halter (1963) set
out as their major objective the development of a model for a
multistage stochastic replacement decision problem, then illus
trated its use by giving consideration to the problem of the re
placement of dairy cows. The work of Jenkins and Halter is
the first study wherein dairy cow replacement was explicitly
formulated as a problem of maximization. Its methodological
constructs were drawn from Bellman's work on dynamic pro
gramming. Its empirical content was particularly valuable to
the study reported in this bulletin.
The second prior major dairy cow replacement study was done
by Giaever (1965) and employed Howard's model as the basic
analytical technique in the determination of optimal replacement
decision policies for two experimental university dairy herds.
Although Giaever's work was very useful in both its method
ological and empirical aspects as a reference for the present
study, three major reservations or limitations regarding his
work were acknowledged. In the first place, the model did not
include consideration of genetic improvement over time. Sec
ondly, the parameters used were only imperfect estimates of
reality, and it was felt they could benefit from additional re
search. Finally, the small number of state variables and the
small number of possible values for each state variable, par
ticularly the variable for prior production, required the ignoring
of information that might otherwise have enhanced the preci
sion of the replacement decision. Another important drawback
to Giaever's work is that it is generally available only on micro
film, and this is an inconvenient reference medium for most
readers.
Many minor references to dairy cow replacement and re
placementrelated reports appear in the literature. Most of
these are concerned principally with predicting production in
the current or next lactation. At the present time, the most
common quantitative guide to culling is the projection of partial
lactation records to the 305day, twicedaily milked, mature
equivalent record. In some instances the cows are also then
ranked on the basis of the deviation of their projected record
from the average for the herd which they are in. Other possible
bases for ranking cows that have been developed are estimated
producing ability (Eastwood, et al., 1966), and most probable
producing ability (Lush, 1937). In addition to these somewhat
more formal cow evaluation methods, there are many "rule of
thumb" guides to culling both recommended and in use. Most
of these are in the nature of a comparison to some standard or
goal. A "normal" cow will produce about onehalf her total
305 day production in the first 120 days of a lactation, for
example, and a cow in her first lactation should produce at about
80%0 of her mature capacity. In general, such guides suggest
that any cow which deviates significantly in an unfavorable
direction from herdmate norms or herd averages in these or
any other of a number of characteristics or performance levels
ought to be further evaluated for possible removal.
The shortcomings of present culling aids or guides are seen
to be in the fact that they do not look far enough forward into
the future, they do not incorporate an adequate number of eco
nomic variables, and they are not definitive as to either if
or whena particular cow ought to be culled. They do not,
furthermore, provide a comparison between the probable conse
quences of keeping a cow now in the herd and those that would
take place if a cow of a specified quality was brought in as a
replacement.
DEVELOPMENT OF REPLACEMENT VARIABLES
AND PARAMETERS
Determinants of Milk Production
Throughout this study, cow performance is measured as
pounds of milk, corrected to a 4% milk fat basis, produced in
the first 305 days of each lactation (305 day, 4% F.C.M.). The
differences among cows in quantities of milk produced are a
function of the complex interaction of numerous genes, environ
mental circumstances, and management factors. One may as
sume that any cow and her potential replacement will be fed
according to the same set of feeding standards. Environmental
changes probably affect all cows similarly, both as to direction
and relative magnitude of effect. Unless one has reason to be
lieve that the present and potential replacement cows will be
treated differently because they are different cows, or that they
will react differently to the same conditions, attention to other
factors is likely to prove more fruitful.
If a factor which differentiates among cows can be measured
and can be related to differences in performance, it warrants
being considered for possible inclusion in a milk production pre
dicting model. A cause and effect relationship need not be
implied, as, for example, production in one lactation does not
"cause" production in the next, though they are highly corre
lated.
Size of Cow
That milk production increases as size increases is. well
documented. As both body weight and milk production follow
about the same pattern of change over the lifetime of a cow,
much of the variation in production due to weight changes is
accounted for if the variable for age is already included. The
younger the cow, however, the more important weight is, rela
tive to age. Johansson (1961) found a correlation of 0.197 be
tween milk fat yield and body weight when age was held con
stant, and a correlation of only 0.08 between yield and age at
calving when weight was held constant.
For milk production at a given level, the smaller the cow
the more efficient she is in terms of energy output relative to
energy input, because the smaller cow requires less energy for
body maintenance. However, larger animals are generally more
efficient (if production is not held constant), particularly within
a given breed, as increases in yield are proportionately greater
than increases in maintenance requirements.
The heritability of body size is relatively high. Mason et al.
(1957) reported heritability coefficients for weight of 0.37, for
heart girth of 0.41, and for wither height of 0.51. Thus, size of
cow can be increased rather easily through selection. In fact,
the authors suggest that in the absence of efficiency measure
ments the best selection basis of cows would be lactation yield,
making some allowance for weight.
Age at First Calving
Age at first calving is much more important in the first
lactation than the following lactations, and becomes less and
less important as the number of lactations increases. Within
the usual one year range of age at first freshening, milk yield
in the first lactation increases with age. From the point of view
of lifetime yields, the earlier first freshenings have an advantage
over the later freshenings, as a greater number of lactations will
ordinarily be completed. Salisbury and Vandemark (1961) sug
gest that if the goal is to maximize production in the first lac
tation, the optimal age for first calving, is about 30 months. If,
however, the goal is maximum economy of production over the
lifetime of the cow, she ought to first freshen at 20 to 24 months
of age.
Age at Calving Each Particular Lactation
The typical lifetime production curve on a lactation to lac
tation basis increases at a decreasing rate until body maturity
is reached at about the third lactation; then it decreases (at
first very slowly) at an increasing rate with advancing age.
This phenomenon is well established and accepted, though the
exact shape of the lifetime lactation curve and its maximum
point will vary from cow to cow and herd to herd. The corre
spondence between age and lactation number is neither perfect
nor uniform, of course. Two cows of the same age but with
differing numbers of lactations completed would be expected to
have different yields. The greater the number of lactations
completed, however, the less pronounced this effect becomes.
Again, weight changes associated with changes in age are gen
erally more important than the age changes themselves. At
the higher ages, however, age begins to become more important
than weight due to the simple effects of aging on the capacities
of the productive processes.
Length of Dry Period
For the individual cow, milk yield increases with the length
of previous dry period for dry periods up to seven or eight
weeks in length. Beyond that there is little beneficial effect on
yield. Salisbury and Vandemark (1961) assert that the optimal
length of dry period is about 55 days. Cows which have un
usually long dry periods, principally the result of low persistency
of yield, tend to be poor producers. The length of the dry period
has fairly high repeatability and heritability. Johansson and
Hansson (1940) reported a repeatability of 0.28 and a herita
bility of 0.32 for this trait.
Length of Calving Interval
Milk yield during a given lactation is affected by both the
calving interval in the prior lactation and the calving interval
in the present lactation. Johansson (1961) cites several esti
mates of the repeatability and heritability of calving interval
that show them to be generally very low. Repeatability is at
about the 0.15 level and heritability essentially zero.
Production in the present lactation is influenced by the length
of the present calving interval through the demands for growth
of the developing calf. The needs of the growing fetus come
into conflict with the urge to produce, and a measurable effect
on production in the latter stages of pregnancy takes place. Most
studies show that these effects are not apparent until after the
fifth month of pregnancy. If a cow is bred back 60 to 90 days
after freshening, as is generally recommended, there would be
no expected decline in production, due to the presence of a calf,
for the first seven to eight months of the present lactation.
In the study by Johansson and Hansson (1940), calving in
tervals of less than 320 days (the average was 390) caused a
decrease of 9.0%o in production in the current lactation, and
3.7% in the next. When the calving interval increased to 450
days, the yield in the current and following lactations both in
creased by 3.5%.
Johansson (1961) concluded that the optimum length of
calving interval seemed to be somewhere between 12 and 14
months, and that it was shorter for cows with low persistency
than for those with high. Salisbury and Vandemark (1961)
stated that a calving interval of 12 months appeared to be opti
mal from the standpoint of lifetime production. Cows with 10
month calving intervals yielded 12% less milk in the combined
second, third, and fourth lactations than did those with 12month
intervals. Cows with intervals of 13 to 16 months yielded about
4% more.
Milk Yields of Close Relatives and HerdMates
In terms of predicting the milk yield of any cow, particularly
those that were "homegrown," one of the best explanatory vari
ables is the average production of the herd from which she
came and into which she will freshen. There are three impor
tant reasons for this. One is that in that herd will be many
relatives, with like genetic capacities. A second is that the levels
and qualities of feeds supplied will be uniform, and nutrition is
the single most important determinant of milk yield. The third
reason why herd average is ordinarily a good predictor is that
the effect of management is essentially constant from cow to
cow in a given herd, and herd average reflects the quality of
that management.
The records of close relatives (dam, granddam, sire's dam,
sisters, and halfsisters) will be useful for predicting levels of
production to the extent that milk yield is heritable. Most esti
mates of heritability of milk yield are in the range of 0.20 to
0.30 (Wing, 1963). For heifers which have not yet freshened,
an estimate of expected yield which is based on the production
of near relatives may be useful. Once the cow has a record of
her own, however, the value of including such information is
reduced materially.
Production by the Same Cow in Prior Lactations
The best single predictor of production in any given lacta
tion is production in the lactation immediately prior to it. The
repeatability of lactation records is generally reported to be
about 0.40 in the United States, and somewhat higher in Europe.
This means that on the average, in the United States, produc
tion in one lactation explains about 16 % of the variation in pro
duction in the next lactation. This does not seem like much, but
it has been consistently shown to be the best single explanatory
variable available.
Correlation coefficients between successive pairs of lactations
are of about the same magnitude. That is, correlations between
the first and second lactations, the second and third, third and
fourth, etc., are approximately equal. The correlations between
lactations generally diminish, however, as they become further
and further removed from each other. Specifically, the corre
lation between the first and fifth is lower than that between the
first and fourth, the first and fourth lower than the first and
third, etc.
Shown below are the simple correlation coefficients between
lactation yields for the 80 Jersey cows that completed five or
more lactations in the University of Florida Experimental herd
which served as the source of basic cow data for the present
study:
Lactation number Lactation number
2 3 4 5
1 .434 .428 .392 .356
2 .363 .407 .306
3 .512 .412
4 .487
The general pattern of the array agrees with what has been re
ported in other studies. Aberrations in this pattern are probably
due to the small number of observations.
One of the most uniform arrays of correlation coefficients
with respect to pattern is one reported by Castle and Searle
(1957). Berry (1945) reported average correlations between
a single record and the next one, between the average of two
preceding records and the next one, between the average of three
preceding records and the next one, etc., as follows:
Records correlated with Correlation
subsequent record coefficient
Single records .35
Average of two records .41
Average of three records .43
Average of four records .44
Average of five records .46
These data show very clearly that beyond combining production
records for the two most recent lactations, little is to be gained
in terms of predictive accuracy by the incorporation of records
of other, earlier lactations.
Miscellaneous and Minor Factors
Clearly, the determinants of production are many and varied.
Sickness, whether chronic or transitory, will affect production.
The same thing is true of injurywhether permanent or tem
porary. Other trauma, such as fright, sudden changes in tem
perature, or the taking away of a calf, may reduce production,
even if only temporarily. A cow may have a nervous disposition
or a placid one. She may be a slow milker, and under certain
herd milking systems or procedures not always be completely
milked out. A cow may be bossy or shy; the amount of feed she
gets may accordingly be above or below optimum levels. The
list could go on, but it would merely belabor the point that the
production of milk is an extremely complex process, and its ac
curate prediction correspondingly difficult.
Empirical Quantification and Selection of
Predicting Variables
Basic Cow Data
The basic cow data used in this study are from the ex
perimental herd of the Agricultural Experiment Station at the
University of Florida. All Jersey cows (for which there were
useable lifetime data) which had left that herd since the early
1940's were included in the sample. Table 1 shows the distri
bution of cows, by number of lactations completed before leaving
the herd, that comprise the basic data set. The variables which
were developed from the basic cow data are presented in Table 2.
.Table 1.Number of cows, by number of lactations completed, in the
basic data set.
Number of cows completing
Number of At least the Exactly the
lactations number of number of
completed lactations indicated lactations indicated
1 369 107
2 262 76
3 186 62
4 124 44
5 80 39
6 41 21
7 20 8
8 12 6
9 6 4
10 2 2
Screening and Choosing Available Variables
Several multiple linear regressions were run using X1 of Table
2 as the dependent variable and various combinations of the re
maining Table 2 items as independent variables. The first run
included all the variables of Table 2, as the primary purpose for
this run was largely one of screening. As subsequent regression
analyses were conducted, variables were eliminated as they re
peatedly failed to reduce unexplained variance to an extent suf
ficient to warrant their inclusion, or until it was realized that
they would not adequately discriminate among individual cows
or between a cow presently in the herd and her potential replace
ment. The variables for age at first calving, length of dry period,
weight of cow, age at present calving, and days carried calf were
discarded on the first criterionthat of failure to significantly
reduce unexplained variance. Herd average was removed because
it did not differentiate among cows or between cows and replace
ments, since these would all be in the same herd, and the same
herd average would apply to every cow which freshened in any
given year.
Analysis of Previous Calving Interval and Prior Production
Only the two independent variables, previous calving interval
and production in prior lactations, remained after the screening
process, and a more extended analysis of these was carried out
with two objectives in mind. The first was to discover the "best"
specification of those variables in terms of predictive power. The
other was to assure that the form of those variables would ulti
mately meet the requirements of the replacement model envi
sioned for use in Phase II of the study.
Previous Calving Interval.The repeatability of length of
calving interval is quite low. The variable screening process
showed that length of previous calving interval is an important
and significant predictor of production when treated as a con
tinuous variable. Since it was desirable to incorporate previous
.calving interval as a discrete variable in the replacement decision
model, the total range of calving intervals in the basic cow data
was divided into three subintervals (Table 3).
That there are differences in proportions between the first
subinterval and the last two is clear. Furthermore, the sub
intervals chosen do discriminate among cows with respect to
expected production, because when previous calving interval was
introduced in discrete form as a shifter in the prediction of pro
Table 2.Variables derived from raw basic cow data.
Unit of
Variable measurement Description
X1 Pounds 305day 4% FCM produced this lac
tation
X2 Pounds Weight of cow (at time of disposal,
after an overnight shrink)
Xs Months Age at first calving
X, Months Age at calving this lactation
X Days Length of previous dry period
Xe Days Length of previous calving interval
X7 Days Time carried calf during the first
305 days this lactation, less 150 (but
always > 0)
Xs Pounds 305 day 4% FCM produced in imme
diately previous lactation
XI Pounds Arithmetic mean of 305 day 4%
FCM produced in all previous lac
tations
Xio Pounds Herd average (computed for each
year that the basic data span based
on all lactations begun the same
year as the lactation of interest)
duction, the coefficients for those shifters were significantly dif
ferent from each other.
The proportions for the eight lactations within each interval
are similar. Graphs were sketched for all three interval classes,
suggesting linear relationships with slight slopes. Simple linear
regressions of proportion on lactation resulted in regression co
efficients of only .015, .011, and .004 for the three intervals, re
spectively.
Calving interval probabilities were thus assumed to be inde
pendent of both lactation number and of production level. The
former assumption seems justified on the basis of the evidence
presented. The latter is justified to the extent that length of
calving interval is related to fertility or breeding efficiency, and
most researchers of this problem show that there is no relation
ship between level of production and reproduction (Boyd, 1967).
Variations in length of calving interval due to purposive acts of
management are always possible, but a prediction of what these
will be is beyond the scope of this investigation.
Table 3.Number of cows in each of three specified lengths of calving intervals.
Minimum Total Length of interval in prior lactation in days
number number <382 382442 >442
of of
of o No. of Proportion No. of Proportion No. of Proportion
lactations cows cows in of cows in of cows in of
completed interval total interval total interval total
2
3
4
5
 6
7
8
9
Grand totals
Proportions
based on
grand totals .468 .252 .280
Production in Prior Lactations.Production in the lactation
immediately prior to the one of interest was found to be the
best single predictor among all the variables evaluated. The
average of production in all prior lactations was a better pre
dictor, but this is actually a combination or composite of at least
two single variables. Aside from using the average, interest
centered on whether and how much knowledge of production in
lactations other than the immediately previous one would help
in the prediction of production in the next lactation. For all
the cows that completed at least six lactations, production in
location 6 was first regressed on production in lactation 5, then
on lactations 4 and 5, then on 5, 4, and 3, etc. Table 4 shows the
combinations of variables for each regression and selected re
sulting statistics.
This analysis showed that little is gained by including pro
duction data from lactations more than twice removed from the
one of interest. The coefficients of determination do not increase
much once the most and second most recent production variables
have been included in the estimating equation.
The next set of regressions included length of previous calv
ing interval as an explanatory variable in addition to production
Table 4.Selected statistics from the regression of production in the
current lactation on production in prior lactations for all cows completing
at least six lactations.
Constant of
Variables regression F
(lactation numbers) (Ibs. milk) value R'
Dependent Independent
6 1,2,3,4,5 3,878 1.4 .18
6 2,3,4,5 3,993 1.8 .18
6 3,4,5 4,978 1.9 .14
6 4,5 4,812 2.9 .14
6 5 6,538 3.1 .08
5 1,2,3,4 1,866 3.3 .28
5 2,3,4 1,907 4.5 .28
5 3,4 2,270 6.8 .27
5 4 3,931 9.4 .20
4 1,2,3 1,341 9.4 .45
4 2,3 1,565 14.2 .44
4 3 2,935 23.0 .38
3 1,2 4,578 5.6 .24
3 2 6,426 4.4 .11
2 1 4,473 9.1 .20
Table 5.Coefficients of determination for regressions of current pro
duction on production in the two immediately prior lactations, with and
without previous calving interval.
Explanatory variables included
Two prior lactations Two prior lactations
Lactation only and previous calving
predicted (from Table 4) interval
3 .24 .28
4 .44 .45
5 .27 .48
6 .14 .23
in the two prior lactations. The coefficients of determination are
as shown in Table 5. An average of about nine points in the R2
was gained, and this would seem to justify the inclusion of both
previous calving interval and production in the two immediately
prior lactations in the estimation equation.
Generalized Production Prediction Model
The generalized production prediction model which was
judged to hold the most promise was one wherein both lacta
tion number and length of calving interval were entered as dis
crete shifters, and production in the two most recent lactations
as continuous variables. Table 6 contains the results from this
model. Prediction of production for the first lactation is not
included. This will be handled as a special case, since none of
the independent variables chosen for the prediction of produc
tion exist prior to the completion of at least one lactation. Lac
tation 2, calving interval 1, serves as the base in the generalized
model. Production in lactation N2 for lactation 2 was arbitrarily
set equal to production in lactation N1 to fill in that gap in
the data. For example, if production in lactation 2, calving in
terval 1 (P2,1) is to be predicted, the expected production is
simply the regression constant (2,142) plus .293 P1 + .380 P2,
(where Pi = P2 any time lactation 2 is being predicted). If
production in lactation 4, calving interval 3 (P4,a) is to be pre
dicted on the other hand, expected production is the coefficient
for the lactation 4 shifter (596) plus the coefficient for the calv
ing interval 3 shifter (1,049), plus the regression constant
(2,142) for a total of 3,787, plus .293 Ps + .380 P2. The complete
set of these constants of production for lactation number and
previous calving interval is shown in Table 7. The production
predicted for any lactation and previous calving interval com
Table 6.Results of the generalized production prediction regression
model for the 40 cows completing at least six lactations.
Significance
Standard of the
Independent deviation Regression regression
variable Mean of mean coefficient t coefficient
(percent)
Lactation 3 1,137 2.42 1.0 2.5
Lactation 4 596 1.27 20 40
Lactation 5 53 .11 >50
Lactation 6 73 .15 >50
Previous calving 738 1.94 5 10
interval 2
Previous calving 1,049 2.95 0.1 0.5
interval 3
Production in 8,510 2,316 .293 4.12 <0.1
lactation N1
Production in 8,397 2,027 .380 4.61 <0.1
lactation N2
Production in 8,624 2,427 
lactation N
Number of observations = 200
Coefficient of determination (R') = .30
Constant of regression = 2,142 pounds
Fvalue for the ANOVA of the regression = 10.2
(this is significant at the 1% level)
bination can thus be thought of as being composed of a fixed
and a variable component. The generalized prediction model per
mits the direct computation of expected production for any cow
for which information as to number of lactations completed, the
length of previous calving interval, and production in the one
or two most recent lactations is available. It can easily be seen
how a chain of predicted production levels can be developed from
an initial starting point, assuming that a knowledge of the prob
abilities of going from one lactationcalving interval combination
to another is at hand.
Reasons for Cow Removals and
Development of Transitional Probabilities
Probabilities of transition from one length of calving interval
to another have already been implied, as it was noted that the
proportions of cows in each of the three designated discrete calv
ing interval classes were independent of lactation number, and
that the repeatability of calving interval from one lactation to
the next is very low. The transitional probabilities therefore
Table 7.Constants of production by lactation and calving interval.
Previous
calving
Constant of production
(the sum of the intercept
and the coefficients of
Lactation interval appropriate shifters
number number from Table 6)
2 1 2,142
2 2 2,880
2 3 3,191
3 1 3,279
3 2 4,017
3 3 4,328
4 1 2,738
4 2 3,476
4 3 3,787
5 1 2,089
5 2 2,827
5 3 3,138
6 1 2,215
6 2 2,953
6 3 3,264
are equivalent to the average proportions of cows in each of the
three interval classes given in Table 3.
Studies of reasons for removal of dairy cows have been
carried out by Asdell (1951), O'Bleness and Van Vleck (1962),
Arave (1962), and Arnold, Becker, and Spurlock (1958). A
comparison of the Florida herd data (Table 8) with these
studies showed that there is an appreciable uniformity in dis
posal reasons in both time and space among various herds. Of
363 removals for nondairy purposes from the Florida herd, 90
(25%) were shown to be involuntary. The remaining 75 % have
been designated as voluntary. It is recognized that these assign
ments are somewhat arbitrary, particularly as they were made
ex post.
In addition to the partitioning of removals into voluntary
involuntary groups for all lactations, probabilities of involun
tary removal were computed on an individual lactation basis.
A plot of these points suggested that a linear relationship be
tween lactation number and probability of failure existed. A
line was then fitted by leastsquares. The resulting equation
was:
Probability of failure = .04212 + .01714L ,
where Ln is the lactation number. Table 9 shows the resulting
probability estimates by lactation number for the first six lac
stations. Estimates from Giaever (1965) and Jenkins and Halter
(1963) are also listed for comparison.
The probability of transition from one lactationcalving inter
val combination to another is the probability of the process being
in any particular calving interval initially, multiplied by the prob
ability of going from any given initial lactation to any other lac
tation of interest. It is possible for a cow to be in any one of six
lactations and three previous calving interval classes at any
point in time, except that if she is in lactation 1, a previous
interval does not exist and she is undifferentiated in that re
spect. The probabilities that a cow will go from any of the 18
conceptually possible states to any of the same 18 are sum
marized in Table 10. This table should be read, by way of
example, as follows: If a cow is in or has just completed her
third lactation (regardless of the length of the previous calving
interval), the probability that she will survive to complete a
fourth lactation, and that the length of the calving interval
previous to the fourth will be two, is .2241. The total prob
ability that she will survive to complete the fourth lactation is
the sum of the three transitional probabilities for going to lac
tation 4, which is .4162 + .2241 + .2490 = .8893. The prob
Table 8.Reasons for disposal of cows in the University of Florida dairy
herd classified by proportions and numbers in the voluntary and involuntary
groups, all lactations combined.
Voluntary Involuntary
Reasons Total no. of disposal disposal
for cows disposed
disposal of for indi Prop. No. Prop. No.
cated reasons of of of of
total cows total cows
Low production 114 1.0 114 0 0
Udder problems 36 0 0 1.0 36
Reproductive troubles 103 1.0 103 0 0
Combinations of the
above three reasons 29 .86 25 .14 4
Old age 9 0 0 1.0 9
Deaths 21 0 0 1.0 21
Other reasons
Diseases 5 0 0 1.0 5
Accidents & injuries 7 0 0 1.0 7
Other, unknown, or
unstated 39 .8 31 .2 8
Totals 363 273 90
Table 9.Probabilities of failure (involuntary removal) of cows, by
lactation number, three different studies.
Present study Jenkins
Giaever and Halter
Lactation Raw Fitted (1965) (1963)
number observations estimates study study
1 .0325 .0593 .0620 .0543
2 .0763 .0764 .0786 .0755
3 .0860 .0935 .0953 .0937
4 .0968 .1107 .1120 .1196
5 .2250 .1278 .1286 .1350
6 .1707 .1450 .1452 .1557
a These figures are based on an analysis of the data which served as the basic data set
for this study. For detail on their computation, see text above.
b The basis for classifying disposals as voluntary or involuntary was similar to that
used in the present study. The figures shown here are a simple average of fitted estimates
from the two herds that Giaever worked with.
c Information on the basis for classification was not given. The values shown are raw
values for from 820 to 987 cows per lactation.
ability that she will fail to complete the fourth lactation, .1107,
is the sum of the three probabilities shown for being replaced
by a cow in lactation 1 for each of the three possible previous
calving intervals. The interpretation of all the other transitional
probabilities is the same. The only possible transition, once
lactation 6 has been completed, is to lactation 1, no matter
whether the decision is to Keep or to Replace. These are also
the probabilities that apply any time the decision is to Replace,
no matter how many lactations a cow has actually completed
at the time of the decision.
Characteristics of the Replacement Cow
Predicted production if any of 15 different cows are kept has
been defined (5 lactations X 3 calving intervals). Production if
the present cow has completed six lactations, or if the present
cow is replaced for any other reason, is now of interest.
The essential question when replacement is considered is
which cow, if any, should be replaced. If all present cows are
compared to a replacement of standard quality, an ordering of
the relative merit of each cow can be accomplished. The quality
of potential replacements is, therefore, assumed to be unrelated
to the characteristics of the present cow.
The question of whether all potential replacement cows ought
to be assumed to be of the same quality remains. If they are
assumed to be of different quality, then it would be necessary
to develop some frequency distribution for whatever qualities
are relevant. To avoid complicating the replacement model un
necessarily, it was felt that a fixed value for quality of replace
ment equal to the mean expected value of the probability distri
bution would suffice. The value of the replacement cow at the
present time was therefore set equal to the mean value of
production in the first lactation (7,945 pounds) of all cows which
completed at least six lactations.
Improvements in the basic genetic capacity of cows to pro
duce milk has significance to the replacement problem, as logic
would suggest that the more rapid this advance, the shorter the
replacement cycle. Lush (1960, p. 702) has this to say about the
amount of genetic advance:
Genetic improvement can approach 1% per year if we con
centrate entirely on production, cull the lowest producers as
fast as we can spare them, and use in natural service only
the sons of the highest producing cows.
Some geneticists estimate a possible genetic change of 11/2 to
2% annually with artificial insemination. In the numerical re
Table 10.Probabilities of transition from one lactation and previous
calving interval combination to another.
Lactation number and
previous calving
interval combination Number of lactation
that will be just completed
completed next
Previous
Lactation calving
number interval 1 2 3 4 5 6
1 1 .0358 .0438 .0518 .0598 .0678 .4680
1 2 .0193 .0236 .0279 .0322 .0365 .2520
1 3 .0214 .0262 .0310 .0358 .0406 .2800
2 1 .4322
2 2 .2327
2 3 .2586
3 1 .4242
3 2 .2284
3 3 .2538
4 1 .4162
4 2 .2241
4 3 .2490
5 1 .4082
5 2 .2198
5 3 .2442
6 1 .4001
6 2 .2155
6 3 .2394
placement models which were run, 1% was used, though the
factor for genetic advance was specified as a variable input to
the models.
Economic Components of the Replacement Problem
The economic components of the replacement environment
are numerous and diverse. Fortunately, costs and returns that
can be assumed to be the same for all cows at a given point in
time can be ignored in the replacement problem. Examples of
these are costs of housing, equipment, land, and most of the
labor.
Seasonal patterns and secular trends in prices, of both fixed
and variable factors, could have a bearing on the optimal re
placement policy. Secular price changes could rather easily be
incorporated into the replacement model which was developed.
Seasonal price differences would be more difficult to accom
modate, but when the assumption is made that a cow presently
in the herd and her potential replacement would both freshen
at essentially the same time, the bulk of the differential seasonal
effects would disappear. The net effect of both seasonal and
secular price patterns would then hinge on the real inherent
differences between present cow and potential replacement.
These are small where the optimal replacement decision is other
wise in doubt. Hence, the possible potential effects of seasonal
and secular price differences on both the expected net returns
and the associated optimal replacement pattern would also be
small. Refinements of these sorts are beyond the scope of the
current study. Attention is more properly focused on the selec
tion and determination of the variable costs and returns which
are of prime importance in the maximization of expected net
returns to the cow process.
Quantification of Variable Costs and Returns
Value of Milk Produced
The quantity of milk expected to be produced by a cow with
any given set of characteristics in any time period is the pro
duction predicted for that cow if she does not fail, less the loss
in production to the process if she does fail. It is assumed that
except for planned replacements, there will be a lapse in pro
duction between the time the cull leaves the herd and a suitable
replacement enters. Though this may be set at any level in the
numerical replacement model, it was entered at the equivalent
of about two months (or onefifth of a 305day lactation). This
may appear high, but since it is weighted by the probability of
failure, it does not amount to a great deal. Expected produc
tion is:
r = y (the probability of failure) (y)
5
where y is the production predicted for the cow if she does not
fail. The value of milk produced is then expected production, r,
multiplied by the price of milk.
Value of Calves Born
The value of calves born is assumed to be a function of:
Their value for veal; their sex; and the production of their dam,
if a heifer. The University of Florida herd produces Jersey
calves which average about 54 pounds each. The probabilities
of a bull or a heifer calf are assumed to be equal at 0.50. Thus,
the expected value for veal of a calf of either sex is .54 times
the price of veal in dollars per hundredweight. This is the total
value of bull calves. Heifer calves are assumed to have value
only for veal if the production of their dam is 2500 pounds or
less. Assuming a veal price of $23.37 (Empire Livestock Mar
keting Cooperative, 1967), the value for veal of bull calves
and heifers from dams producing 2,500 pounds or less is .54
($23.37) = $12.62. Heifers from dams producing at 14,500
pounds or more are assumed to be worth the same $12.62 for
veal, and an additional $62.38 for milk, for a total of $75.00.
Values for heifers from dams of intermediate production levels
are proportionate linear interpolations of the range of values
from $0 to $62.38. Thus, the expected total value of a calf of
unknown sex from a dam of production level r at a given veal
price is:
(the probability of a bull calf) .54 (veal price)
plus
(the probability of a heifer calf) .54 (veal price)
plus
(the probability of a heifer calf) 12.99583*" r 1)
\2500 /
*12.99583 = $62.38
14,500 1)
S2,500 /
By substituting 0.5 for the probabilities of bull and heifer calves
and combining terms, the value of calves born becomes:
.54 (veal price) + 6.49791 1.
S2500
Salvage Value of the Cow
The salvage value of a cow is in part her value for milk pro
duction, and in part her value for beef. The average disposal
weight of the 40 Jerseys completing six lactations in the Uni
versity of Florida herd was 921 pounds. Table 11 shows the
average ages of those 40 cows at each lactation, a weight index
based on age, and estimated weights by lactation based on the
average disposal weight and the weight index. The beef, milk,
and total values that were developed for cows with different
numbers of lactations are shown in Table 12. In addition to the
weights already developed, a cow's value for beef depends on
its condition. Condition is reflected in the different carcass
grades and associated prices shown in Table 12.
If the total value of a "close" heifer is $275, and its value
for beef is $144, then its value for milk production is $141. The
value for milk production of a cow which has completed six lac
tations was earlier said to be zero. Its total value is then $164.
The value for milk of cows having completed a number of lac
tations intermediate to none and six diminishes by onesixth
of the $131 or about $22 for each additional lactation completed.
Depreciation in Value of Cow
The difference in salvage values of a cow at the beginning
and at the end of any given lactation is defined as the depre
ciation in the value of that cow for that lactation.
Interest on Capital Value of Cow
Interest on the capital value of the cow is the opportunity
cost of that capital. The capital value of the cow is assumed to
be her salvage value at the time she begins a new lactation.
Feed Costs
Total feed requirements are the sum of the cow's nutrient
needs for maintenance, growth, reproduction, and production. It
SThis expression is not generalized to other breeds, herds, or basic price
relationships. If other replacement models were to be developed, it would
be well to incorporate an expression which would be generalized in those
variables.
Table 11.Average ages and weights of the 40 Jersey cows completing
six lactations in the University of Florida herd.
Age Index of Estimated Assumed
Lactation (months) weight weight weight
1 30 1.200 768 767
2 45 1.090 845 850
3 60 1.020 903 917
4 75 1.000 921 917
5 90 1.005 916 917
6 105 1.010 912 917
a Source: U. S. Department of Agriculture (1967a).
b. Developedfrom the application of the weight index to the average disposal weight
of 921 pounds. The index was applied by dividing the average disposal weight by the index
indicated for each particular lactation. For lactation 1, for example, 921 was divided by
1.200 to yield the 768 shown under "estimated weight" for that lactation.
e These weights were assumed for purposes of computing feed requirements in the
section titled "Feed Costs."
is assumed that a cow reaches her maximum weight by the time
she has completed two lactations, and that her weight remains
constant from then on. The nutrient components of major in
terest are the digestible protein and energy levels.
The average length of a calving interval among the 40 cows
that completed six lactations in the base herd was 418 days. The
weights assumed for these cows in computing feed requirements
were given in Table 11. Feed requirements on a lactation basis
were developed based on Morrison and Morrison (1958, p. 1087),
the weights of the cows, and their production. These total re
quirements are shown in Table 13, including a breakdown be
tween the fixed and variable portions. A computed total, which
reflects all feed needs reduced to a single, allpurpose, one
specification ration is also shown. Details as to the derivation
of these total feed needs may be found in Smith (1968).
The allpurpose ration contains 74 therms and 10 pounds of
digestible protein per 100 pounds of feed. Pounds of the 7410
feed required, by components, by lactations, are:
Lactation number Fixed component Variable component
1 3,993 .4488r
2 3,707 .4488r
36 3,364 .4488r
where r is the expected production of the cow as defined in the
section "Value of Milk Produced."
The cost of feed is the quantity of feed required multiplied
Table 12.Carcass grades, prices, and salvage values of study cows, by lactation.
Index Value of Cow for
Carcass Beef of Cow
Lactation grade price price weights Beef Milk Total
(dols./cwt.) (pounds) (dols.)
"Close" Standard
heifer or common 21.00 1.075 687 144 131 275
1 Low standard
or common 20.50 1.05 768 157 109 266
2 High utility 20.00 1.025 845 169 87 256
3 Utility 19.50 1.0 903 176 65 241
4 Low utility 19.00 .975 921 175 43 218
5 High cutter 18.50 .95 916 169 21 190
6 Cutter 18.00 .925 912 164 0 164
a Source: Empire Livestock Marketing Cooperative (1967).
Table 13.Actual and assumed digestible protein and net energy requirements, by lactation, for the 40 sixlactation
cows in the basic data set.
Nutrient requirements 418 day lactation
Lactation Actual Computed
Fixeda Variableb Total total
D.P. N.E. D.P. N.E. D.P. N.E. D.P. N.E.
1 384 3,069 389 2,384 773 5,453 756 5,594
2 338 2,985 396 2,422 734 5,407 733 5,425
3 292 2,818 442 2,704 734 5,522 741 5,483
4 292 2,818 441 2,699 733 5,517 740 5,477
5 292 2,818 418 2,557 710 5,375 719 5,320
6 292 2,818 417 2,554 709 5,372 718 5,317
Totals 1,890 17,326 2,503 15,320 4,393 32,646 4,407 32,616
a Based on average assumed weight of cow, with an allowance for growth and pregnancy as shown in Morrison and Morrison (1958).
b Based on the average production for the lactation at .049 pounds digestible protein and .30 therms per pound of 4% FCM. Average pro
duction of milk for lactations one through six was 7,945, 8,074, 9,015, 8,996, 8,552, and 8,513 pounds, respectively.
c Developed as described in detail in Smith (1968).
by its price. As this price will vary in both time and space, it
is specified as a variable input to the replacement model in the
following sections.
Net Variable Returns
In the computation of net returns in the replacement model,
salvage values, depreciation, and interest were incorporated in
the value of the cow (state) which characterized the process
at the beginning of a lactation (stage), and the one which
characterized it at the end. The replacement model nominally
requires that the manager buy into the process at the beginning
of each stage, and that he sell out again at the end of that stage.
The differences between these purchase and selling prices in
corporate charges for interest and reflect changes in salvage
values (depreciation) as well.
Table 14 is a list of major components in the computation
of net returns. Only the present values and expected future
salvage values need further elaboration.
Present salvage values are a function of beef prices, cow
weights, and heifer prices. The beef price (variable input) is
first multiplied by a constant for each lactation which is the
product of the beef price index and cow weight (see Table 12).
This gives the value of the cow for beef. The cow's value for
milk is the heifer price minus the value for beef, scaled down
by onesixth for each succeeding lactation. In the case where
a cow has completed six lactations and must be replaced, or the
decision is to replace her, the value of the process is simply
the heifer price plus a transaction cost. This transaction cost
is the sum of all sales commissions, delivery or hauling costs,
and other such charges not normally included in the sale price
of the animal. All present salvage values are multiplied by
1 + x, where x represents the opportunity rate of return for
the capital that will be invested in the cow over the forthcoming
lactation period.
The values of the cow at the end of the forthcoming lacta
tion period are the same, lactation by lactation, as those at the
beginning, except that they are discounted by the factor 1 minus
(probability that the cow will fail) x (probability of failure due
to death). In the event of death, the cow is considered to have
a salvage value of zero. The discount factors thus derived pre
multiply the overall values of the cows based on their values for
beef and milk to determine the expected worth of the process
when the manager, in a figurative sense, "sells out" at the end
of a productive period. The factor (1 + x) is used as a pre
multiplier to determine the cost of "buying into" the process at
the beginning of a production period. Thus, in Table 14 the
former is shown as a positive return, the latter as negative.
In Table 14 prices for feed, beef, milk, heifers, and veal, as
well as transaction costs and interest rates, are shown to be
variable within appropriate market limits. Feed and milk prices
are in dollars per pound, beef and veal prices in dollars per
hundredweight, heifer prices and transaction costs in dollars, and
interest rate in the decimal equivalent of the appropriate market
rate percentage.
DAIRY COW REPLACEMENT MODEL SPECIFICATIONS
General Specifications
The model of the dairy cow replacement decisionmaking en
vironment determined to be most appropriate to the present
study is a "decisiondetermined stochastic Markov process with
economic rewards." A Markov process is simply defined as a
stochastic process wherein the probability distribution of out
comes at any given stage depends only on the outcome at the
previous stage. The term "stochastic process," in turn, is used
to refer to any process which is made up of a series or sequence
of experiments or observations wherein the outcome at one
stage depends on the outcome of the previous stages in a prob
abilistic sense. If a Markov process is decisiondetermined, then
the probability distribution of outcomes (states) is a function
of the decision which is made. Associated with each state is a
return or reward which may be specified in physical, economic,
or other terms appropriate to the particular problem or process
being modeled. For further information on Markov processes
and associated concepts and techniques, Kemeny and Snell
(1960) is suggested.
A "process" in the present study is some present or initial
cow, all her replacements, and all the replacements' replace
ments. When a planned replacement takes place, the removal of
the old cow and the entry of the new will be accomplished simul
taneously. When an unplanned replacement takes place because
of the failure of the cow which is kept, a lapse in production of
two months is assumed.
A "planning horizon" is defined as the number of stages over
which the process is to be evaluated. In the dairy cow replace
ment problem as developed in this study, the length of each stage
Table 14.Major components in the computation of net returns in the dairy cow replacement problem.a
() Feed costs
Lactation
just begun Feed costs
1 (3993 + .4488 r) (feed price)
2 (3707 + .4488 r) (feed price)
36 (3364 + .4488 r) (feed price)
() Present salvage value plus interest on that value:
Lactation
just Present value of the process
finished
1 (1 + x)
2
3
(1 +x)
(1+ x)
beef
[(price) 8.064 +
8.661 +
9.030 +
8.980 +
beef
[(price) 8.702 +
[(heifer price +
heifer beef
.83333 1 price [7.385 (price)]
.66667
.50000
.33333
heifer beef
.16667 J price [7.385 (price)]
transaction costs) 1.05926]
+ .07640 (heifer
+ .0'9354 (price
+ .11068 ( +
+ .12782 (trans.
+ .14496 (cost
(+) Expected future salvage value less losses from death:
Lactation
just begun Expected future value of the process
beef heifer beef
1 .98518 [(price) 8.064 + .83333 price [7.385 (price)] !]
2 .98090 8.661 + .66667
3 .97662 9.030 + .50000
4 .97233 8.980 + .33333
beef heifer beef
5 .96804 [(price) 8.702 + .16667 price [7.385 (price)] []
beef
6 .96376 (price) 8.436
(+) Value of milk produced = (milk price) (r)
r
(+) Value of calves born = .540 (veal price) 6.49791 2500 1
a See accompanying text for detail as to the specification of the variables and the derivations of the constants.
1=1 a See accompanying text for detail as to the specification of the variables and the derivations of the constants.
is a stochastic variable, so the total length of the planning hori
zon is only probabilistically related to the number of stages.
The "evaluation" of the process which has been referred to
is the comparison of the effects that alternative decisions have
on the level of rewards which are generated by the process over
the entire planning horizon. Since these rewards are forthcom
ing at different points in time, they must be appropriately
discounted for either or both the opportunity costs and time
preferences for the reward. The evaluation then consists of the
selection of those KeepReplace decisions which maximize the
expected present value of the reward. This study designates
milk the reward in one instance and net returns in the other.
The Dynamic Programming Replacement Model
The dynamic programming approach to the solution of prob
lems of optimization is relatively new. Applications of dynamic
programming techniques are expanding at a rapid pace, though
there remain few in agriculture. A brief and understandable
introduction to the principal assumptions and methods of dy
namic programming is contained in the article by Burt and
Allison (1963). A longer expository paper on dynamic program
ming was written by Howard (1966). For complete detail as
to the way in which the dairy cow replacement problem was
structured to conform to dynamic programming concepts and
principles, see Smith (1968).
The computational method or approach of dynamic program
ming is its most unique characteristic. It begins at the final
stage in the planning horizon and proceeds backward in time,
stage by stage, until the present time is reached. If there is
just one stage left in the cow replacement process, returns to
the process will be realized during that stage, and possibly at
the end of that stage in the form of salvage values. Decisions
are selected so as to maximize the sum of these two sources of
returns. Equation (1) is the Markovian dynamic programming
formulation of the dairy cow replacement problem when there
is just one stage remaining. This formulation is similar to
that of Howard (1960), and his work should be consulted both
for additional detail and generality. In equation (1), fi(1) is the
maximum present value of expected net returns (or production)
to the cow process if there is one stage remaining and the process
is in state i. The maximization is made over the decision d which
can take on the values Keep and Replace. Given a state i, and
a decision d, there is a probability p, and an associated return r,
for each state j that the process can be transformed into. The
products of p and r are summed over all j's to get the expected
returns for the single remaining stage. To this is added the
discounted expected salvage value of the process at the end of
the planning horizon.
The backward, recursive relationship that is established for
planning horizons of any length can be seen clearly in the com
plete system of equations (1) through (5), as follows:
f m fd(0)
fi(1) = max [ pj rij() + Pij j() ] (1)
Z[ d d Z d
fi(n2)= max [ Pij rij(n) +j Pij fj() (2)
d
fi() = max [ j Prij(n) j j( ] (3)
d
fi(N) max [ pij ij(N) + Pijj(N1) (5)
n = 1,2,... N = the number of the stage the process is
now in,
i = 1, 2,..., I = the state of the e e process at the beginning
of a stage,
j = 1, 2,..., J = the state of the process at the end of a
stage, and
d = k if the decision is to Keep, r if the decision is to Replace.
Net milk production is:
d = g (6)
ri,n ij,n gij,n (
where
yi = production if the process succeeds, and
gi = loss in production if the process fails.
Production if the decision is to Replace and the replacement cow
does not fail is:
ij,n = Yij,n ,n. (7)
Yn = REPVAL [1 + GENPC (Nn) ] (8)
where
REPVAL = production of the replacement cow in the first
stage, and
GENPC = stage by stage change in the basic genetic capacity
of the cow population to produce milk.
Incorporating genetic progress (obsolescence) into the model
in this way makes each different length of planning horizon a dif
ferent problem, for the solution for the fifth stage of, say a 10
stage problem, is not the same as the solution for the fifth stage of
a fivestage problem. This is because the production of the re
placement cow at the fifth stage for the two problems is different.
It is different at every other stage of the same number as well.
The probable loss in production due to failure of the replace
ment cow is:
rijn = pF Yr.n (9)
PENALTY
where
PFis the jth element in a vector of the probabilities that the
state into which the process is transformed will fail at any stage.
These are the cow failure probabilities that were fitted from the
basic data of the present study as shown in Table 9.
The variable PENALTY is the reciprocal of the fraction of a
lactation which is lost during the time it takes to replace an un
planned (involuntary) removal.
Production if the decision is to Keep and the cow does not
fail is:
rA,n= n ij ,n (10)
yikn ci1 b9n+2
Ay n = c + an+l1 by+2 (11)
where
cij is an element in a vector of the constants for production
shown in Table 7, and yn+l and yn+2 are production in the two
immediately previous lactations. The y's are subscripted n+1 and
n+2 rather than n1 and n2, so that they will correspond to the
backward approach of dynamic programming. The regression co
efficients for the two prior production variables are a and b as
shown in Table 6.
A special case results when the process would progress to the
second lactation under the decision to Keep, as there is no lacta
tion datum for that cow for stage n2. The variable yn2 was sim
ply arbitrarily set equal to yj,n1 to fill in the data gap.
The probable loss in production due to the failure of the cow
kept is:
g, = pFy (12)
PENALTY
computed on the same basis as shown for gin in equation (9)
above.
Equations (7) through (12) are generalized as to the state of
the process and the number of the stage. The state of the process
at any stage is a function of the state at the preceding stage and
the decision made there. These factors determine both the states
into which the process may be transformed given some present
state of the process, and the probabilities of each of those trans
formations taking place. This introduces an additional stochastic
feature which seemed necessary to a meaningful and useful repre
sentation of the cow replacement decisionmaking environment.
Let:
p = the probability of transition from state i to (13)
state j for the decision Replace
where
J
P = 1.0, and 0 pij 1,
j=1
and
k = the probability of transition from state i to (14)
i* state j for the decision Keep
where
J
Z p= 1.0, and 0 pij 5 1.
j=1
Expected production for a given present state of the process,
under decision d, and at state n, is then:
J
r p rn. (15)
j=1l
Under either decision d, many of the elements pij and rij
are zero, since for any given state of the process' there are re
strictions with respect to the transformations which may take
place.
The discount factor is:
3 = (1 + x) 1 (16)
where
x = the stage discount rate.
The discount factor, P, is not subscripted because the ex
pected return in all remaining stages at any given stage is
accumulated and discounted again at each stage. For example,
at the first stage (when numbered backward in time) the ex
pected optimum return at stage 0the salvage valueis dis
counted once in arriving at the optimum return at stage 1. At
stage 2, the expected optimum return at stage 1 is also dis
counted once. This in fact discounts the return at stage 0 for
the second time, as the return had already been once discounted
in the computation of the optimum return for stage 1.
The general recurrence relation of equation (3) was solved
by what Howard (1960) terms the valueiterative method.4 The
iterations proceed by computing and storing the optimal returns
for each possible state with one stage remaining. Then the
optimal return with two stages remaining is found by selecting
the d for each possible state of the process at stage 2 that maxi
mizes the sum of the returns at stage 2 and the discounted opti
mal returns for the state at stage 1 that results from the decision
at stage 2. The computational process is repeated for the third,
fourth, etc. stage until the number of iterations equals the num
ber of stages in the planning horizon. Bellman (1957) shows
that a constant decision policy emerges as the number of stages
increases under the valueiterative method. Once convergence
has taken place, the associated policy is then also optimal for
any greater number of stages. A formal test for convergence
is suggested by Burt and Allison (1963) and by Giaever (1965),
but was not carried out in the present study.
Method of Incorporating Production in Prior Lactations
The variables for production in the two prior lactations are
given to the nearest pound of 4% fatcorrected milk. In order
to meet the Markov requirement of independence between stages,
it is necessary to know or assume values of each prior production
variable at each stage. That is, the state of the process at any
stage must be unambiguously described (in advance) in terms
of a specific lactation number, prior calving interval, and level
of production in the two prior lactations.
The number of lactations and the number of previous calving
intervals were earlier set at six and three, respectively. To re
duce the number of possible states to a level which was thought
to be an acceptable compromise between precision and costs of
computation, the range of possible production levels for prior
'Howard also describes the policyiterative solution routine for planning
horizons of infinite length. In addition to the value and policyiterative
methods, the replacement problem may be transformed into a linear pro
gramming format and solved by the ordinary Simplex procedure. For
information regarding the relationship between the linear programming
and dynamic programming problems, Hadley (1964), and Kislev and Amiad
(1968) are recommended.
lactations was set at 5,000 to 12,000 pounds. This range was
then divided into intervals of 250 pounds. Production data were
rounded to the nearest of the 29 production classes thus created.
With 29 values of the state variable for production in each prior
lactation, and 18 lactationcalving interval combinations, there
were 29 X 29 X 18 = 15,138 states that the process could be in
at any stage.
The question arose as to the nature of the transition between
some combination of production class intervals in the two prior
lactations at one stage, and all possible combinations of produc
tion class intervals in the two lactations prior to the next stage.
Production in the lactation once removed at a given stage is the
same as production in the lactation twice removed in the next
stage, because they refer to the same lactation. Production in
the current stage is stochastically related to production in the
two prior stages. If the probabilities of transition from every
possible combination of production class intervals in the two
prior lactations to every possible interval for production in the
current lactation were to be incorporated into the replacement
model, the solution to the replacement problem might be im
proved, but the costs of computation might be prohibitive. The
difficulties of programming the transitional probabilities into the
replacement model, and the additional costs of computation, were
felt to outweigh the probable returns in terms of better replace
ment decisions. The direct relationships between production in
prior lactations and production in the current lactation were
therefore assumed to be deterministic.
EMPIRICAL ANALYSES OF THE REPLACEMENT MODEL
General Comments
The replacement model was run for planning horizons of 10,
15, and 20 stages, using the maximization of net returns as the
objective criterion, and a representative set of variable prices.
A comparison of the results for the 10 and 15stage planning
horizons revealed that the optimal replacement decision was
different for only eight of the 15,138 possible states of the pro
cess. When the results for the 15 and 20stage horizons were
compared, no differences in the optimal replacement policy were
encountered. This suggested that convergence to a constant
policy was near, though one can never be certain that complete
convergence has been obtained without resorting to more elab
orate tests.
Five numerical runs of the 15stage model were then carried
out. The first four of these had as their objective criterion the
maximization of the present value of expected net returns. The
combinations of variable price inputs used in these runs are
shown in Table 15. The fifth run of the model had as its objec
tive the maximization of the present value of expected produc
tion. The nonprice variable inputs were the same fodr all five
runs.
Table 15.Variable price inputs to the net returns runs of the dynamic
programming replacement model.
Run number
Variable input 1 2 3 4
Feed cost 0.04 0.032 0.04 0.04
Beef price 20.0 20.0 20.0 24.0
Heifer price 275.0 275.0 275.0 303.8
Veal price 23.5 23.5 23.5 28.2
Milk price 0.065 0.065 0.052 0.065
Transaction cost 15.0 15.0 15.0 15.0
The value of the Keep decision minus the value of the Replace
decision was printed out for all states for all five problems. If
this figure was negative, it indicated that the optimal decision
was to Replace. If positive, then the optimal decision was to
Keep. The absolute values of these figures were the "costs of
the wrong decision."
Figure 1 is a photographic reduction of the computer output
for lactation 3, calving interval 1, for the first net returns run.
The output for other lactationcalving interval combinations and
for other runs are reproduced in Appendix D to Smith (1968).
Small vertical lines are drawn in to separate the values of yn. 2
for which, for a given value of yn. 1, the optimal decision would
be Replace from those for which the optimal decision would be
Keep. The most costly wrong decision for a lactation 3, calving
interval 1 cow was associated with the state where the n + 1 and
Yn+2 values are both equal to 12,000 pounds. The cost of this
wrong decision would be $191.71. The least costly wrong de
cision would occur where y, +1 equals 7,500 pounds and y,+2
equals 9,500 pounds, or where y, equals 7,750 pounds and
Yn 2 equals 9,250 pounds. The cost of a wrong decision at either
of these states is actually zero, and indicates that, other things
being equal, the dairyman ought to be indifferent as to whether
he keeps or replaces cows with those particular characteristics.
Value 6037 0DAI 14 o_ 4o3.7 73 L_
o MATRX OF CEP OCISC VLUES FDR SGE s
CLLL 07 065 u 2__12L.
305.56 103.00 67.7_O9.2 67._1 e0 _5. 40.05 76.05 _72,75 ,<>65 60_55 335o.44 57oS4 Z.,4.34l*_
312.46 67.30 r.30 64.20 06.09 7S.66 73.70 77.79 736 60.59 65.99 63.37 57.29 53.304 9 44.
5,250 40.7" 3a.7 32.77 2..57 24.47 20.37 36.27 32.3 7O. 1.9 0.14 .' .34 I 2.4
99.24 95.t4 1. 02 t.3 7P2.P3 7 .7.3 74.63 70.51 66.42 62.32 57.27 50.12 5S.P7 45. 0 43.*7
5.500 3.73 33. 33.. 1 03 2.4 21.3. 37.21 33.10 9.06 4.00 o.0o 3.30 7.47 33.59 I36.5
6.00 .JL 27.2 23.l; 19.08 14.0i 10.88 6.78 2.64] 1.42 5.52 9.63 13.73 17.01 21.9
e9.7r 2t.6 73.6 77.44 73.  65. b4 61.06 56.00 52.64 .736.7 .e 40.5 9.076 17.3T
6,250 23.aJ 24.13 0o.7 Ie.' 3.a72 7.72 3.52 0.4 .5.s .6o 12.79 1t..9 7 19 0 7.
ee.5a U2.l4 a.oJP 7,.2e ,o..u o_.o7 st.is 57.50 5'.77 ,+.h7 45.7 s.t.4 37.O7 TT.. 79.17
6.50 0as.  .906 .3 3I.73 2.06 4.66 0.451 J.3.5 7.75 113.5 15.195 20.05 264.3S .?6
73.02i 73.3. . 3.3 67.32 62.92 5.0ll 54.71 50.63 0.51 42.41 38.11 3.21 37.10 7.
71.26 70.et 72.06 67.6 63.86 .9.75 55.65 _51.55 47.65 43.35 3.r25 15.35 31.00 51.94 27.0
7,000 e1.74 6.04 a0 54 .3 2.33 1.77 5.97 9.7 14.07 I3.65 22.75 27.97 37.7 1.1I
77.0 73.00 ee.60 64.79 40'.O 56.59 52.9 9.3 .29 40.14 1.07 077. 21.79 190.
7,250 ) l._e 11. 7 7.37 J.271 0.03 5.22 9.33 3.1 1 19.n4 22.71 27. 37.5 371 .6 1.9
7.04. 70.04 (4 .73 01.63 57.5 '.3 3 33 5.2341 5  2 37.02 32.92 2.~ 2.72 ?7. 
7,500 32.43 6.33 4.10 3.o00T 5.22 9.32 13..2 39.66 22.7 26.94 32.72 36.2 03.04 3'.3t
.7 . 02.7 57.67 50.07 50.27 06.36 2.06 37.9 3.96 29.76 5.6 2.56 317. 173.15
7,750 .2.. 4. 00 430 4 .2 13.7?2 .o1 23.23 77.34 33.55 37.6 463.0 6'.39 .044
67.i 43.51 59.01 55.31 51.231 4 7. IC 43.00 3.10. 34.O_ 9.70 26.60 2_.5 1 .4 '. ".3
8,000 J.0l 0.rs 4.40 .65 13.75 0. 17 24.07oi 20.17 39.50 37.79 62.O 4o.30 50.09 59.96
8,2060.556 2 . 4. 6 2 39 1. 1
63.24 7.316 73.07E 40.98 44.e 07.77_ 36.77 03t.67 .26. 22.5 .I1.5 13.1 4 9 .06 31 0 .
8,500 5.0a 11.62 3I.72 30.a0 26.3j 30.41 34.i2 60.74 04.B4 51.43 55.53 59.169 66.3 79.I'
e.12 5.02 o9.01 44.90 40 .0 35.77 31.67 27.57 22.66 ; I 96 12.79 S.19 4.09 j 0 6.I
8.750 2.72 36.e73__.l 27.15 31.25 353536 43.7 0s.97 52.54 56.64 o6.74 67.5 71.55 75.65
54.04 4.4 44.91 3O.Pi s7.7 731 .69 27.5 2 .72_ 17.62 3 .11 1.04 1.07 1.74 _7 4
9.000 017.66 .t.7t 2e. 3 6 .5 36.764 2. 47.06 51.19 57.a7 61.58 6.9l 74.0 7.0An "7.95
9 .91 40.02 00.3 35.6 33.60 26.90 70.67 316.57 2.47 6.25 2.3 1 .' 0.07 17.59 19.0
91,750 2J.17 27.27 J3.6 37.e0 4J3.81 7.92 52.02 60.55 64.66 73.40 77.50 0 5 .I60 0.14 "4.4
40.37 4.OI '5.67 7o.7 25.65 30.60 15.7 11 .6 5.33 1.01 43.09 9.5 ~ 33.009 30.9 '97
9,500 j.1 i _.J 0L_ 7,L _4,24__ .___1 6 11 4 02.95 61.25 7 1. 15 1P9 3 .9 9 4. 7 6q.e3 1.4L __
3 .22 :.1. 2.23 .J 1. _J 11,_, 14 L lZ.. __ l .._ LL. 1 2uL .7 12.2.04 5,14 2",
9,750 _1.67 0o.7 44.07 n2.7 5 .o5 65.91 70.03 14. 11 38 507. 02.07 1310.2 t305. 34.6
3~.1 2 7.1 22.7813 3.73 5 3 9.39 7.T1 .9 5.03 114.56 15.6 t.76 26.*0 30.70 91 .
10,000 4,2.0 7.0. 55.3 5,5 07 63 94240 .1.5A 6.. 909 7 0 7,00 7710.3 Lo 1.07 __2. 15i__
i.,Q0B 22,. 30.90 3.95 B.25. 9.151_ .34 7 .**4 1?2.7 16P.90 9 .__2.9Z.07 31 4.56 ." .6
10,250 49.76 s9.0P (2.9 67T.09 75.82 79.92 09.24 93.34 97.*4 100 .0 112.14 116.25 175.57 12.067
.4 11.74 3. 7.4 3.004 3.7 7.27 15.0 19t.28 23.1 32.55 36. 4n.7i a.s 5.q
10.500 _6.a 6 .07,3__ 66.47 _7_7._ .a9___ 6.99 9.01 03 IOLL .10.4* 1H.5L 11 .6*tlZ.593_13, 3_196.74_ _
L.Att lC*._ 1.31 _2,L_1 f ._ .7? .. 19. 22.76 2 6.96 _,494 39.0 __A3L__ 123 _95.33 1**
10,750 6s.35 72. 2 t.23 6.3 90. 9.0 03IOJ.51 113.40 117.50 121.60 1312.40 39.6o0 309.60 4i11
11.70 3.0 3 7.e7 12.97 311.07 26.13 21.25 30.35 03.76 056.0 56.9 59.99 63.19 71.,75
11.000 70.6 3_.60_ 3.7u_0 92.93 103.3311 07..3 3_131 .53 121.37 _325.47_ 313.7 37._9__3009 2971 3. 9__ 35_
1 0.622 [3.30 7.2 _._6 l9,<6 ?'.! 9__3 .9' _9.75 44 92_.__',0 5?A__ 3. _&L.4 _Z. L5 S.?5
11,250 2.35 92.73 t 6.61 106.23 110.34 13.64 1231.76 327.36 131.96 141.96 145.96 5t.77 10.06 360.76
5.57 / 9.1 Ii.*7 22.67 33.02 35.13 30.23 07.33 51.61 55.51 64.35 67.45 7B.30 7.240 I.4
~910_lloll),500 95.20 99.36 303.0_ 3. 360 321.0 136,0 145.7219.L L59 _.t9 L5._7.^_L2t.95l7.25.
13.35_ 21.2' 25__3.3 20.057 47. 3._ 50.,?__ 3,I 1. 54.23 .50.13 47.69 _71,7?9n0,5, 1_S l.t _9,' ",l'
11,750 102.27 106.37 117.17 121.27 130.59 34.69 30P.79 340.t2 152.22 a56.32 o67.37 173.77 53.33l 15. ?
3.e0 ;7.32 3l.F0 41.00 45.10 03.27 57.39 61.49 70.00 74.13 78.28 77.70 91.70 10'.01 105.31
Value 5,000 5,250 5,500 5,750 6,000 6,250 6.500 6,750 7,000 7,250 7,500 7,750 8,000 8,250 9,500
of ,750 9,000 9250 9,500 9,750 10,000 10,250 10,500 10,750 11,000 11,003 11.250 11,500 11,750 12.000
Figure 1.Sample computer output of the DP model.
Under the decision Replace, the present value of expected
net returns for run one is shown to be $2,153.92 (designated
cull return, Figure 1). For runs two through four they were:
$2,743.03, $1,065.29, and $2,141.85, respectively. These returns
are the same no matter what the initial state of the process
that is, the return to the Replace decision at any stage is inde
pendent of the state of the process at that stage. This is because
replacement cows are assumed to be the same at any given stage,
and their quality is not related to the quality of the cow being
replaced.
Summary and Comparison of the Results
of the Net Returns Runs
Without exception, the maximum level of y+ 2 (production
in the next to most recent lactation) at which replacement would
take place under the optimal policy remained the same or de
creased as the length of the calving interval increased for a
given lactation and a given level of y:+ 1. In other words, the
rate of replacement diminished as the length of calving interval
increased, other things being equal. This effect was consistent
with expectations.
For a given calving interval and yf +1 level, the maximum
Y + 2 level at which replacement took place increased or remained
the same as the lactation number increased, except in the fifth
lactation, where it dropped off about one production class inter
val (250 pounds). This indicated that the rate of replacement
generally increased as lactation number increased, other things
being equal. The exception which showed up for the fifth lacta
tion is seen as an aberration in the basic data, for the overall
effect is in line with what experience would lead one to expect.
The prices used for run one are considered to be at the
"base" or "representative" level in the present study. The 20%
reduction in feed prices in run two brought about a 27 % increase
in net returns for the Replace decision. In run three, the 20%
decrease in milk prices resulted in a 50% reduction in net re
turns. These results definitely are what one would expect as to
direction, and entirely in order as to magnitude. The effects of
the 20% increase in beef and veal prices which were imposed
in run four resulted in a slight reduction in net returns. On first
thought this might seem inconsistent with expectations. When
it is recognized, however, that over the long run beef prices are
probably more important to the cost side than they are to the
revenue side of net returns, the observed effect of beef price
changes is logical.
A comparison of the optimal replacement policies for runs
one and two revealed that where there was a difference it was
in the direction of earlier replacement. That is, as feed prices
declined, cows were replaced at somewhat higher production
levels. These differences were of consistent direction, and
diminished slightly as length of calving interval and lactation
number increased. The direction of the effects of a 20% reduc
tion in the price of milk on the optimal replacement policy was
mixed, though of about the same overall magnitude as that of
the feed price decrease of run two. Although the smallest dif
ferences in present values occurred with changes in beef prices,
it is here that the greatest differences in optimal replacement
policies were encountered. Run four, when compared to run one,
consistently showed the optimal policy to require earlier re
placement when beef and veal prices increased by 20%. Table
15 also shows the price of replacement heifers to have increased
by about 10%, since about half the price of heifers is composed
of their value for beef.
Based on these comparisons of the results of the net returns
runs, it seems valid to assert that even moderate changes in the
relative level of important input or output factors will not alter
the optimal replacement policy to an economically significant
degree when compared to the "base" price situation. Thus, once
a solution to the replacement problem is found for some repre
sentative set of factor prices, it will probably not be rewarding
to solve it again simply because one or more factor prices have
changed, even if by fairly large amounts.
Results of the Production Run with Comparisons
to the Net Returns Runs
A single run of the DP model was executed wherein the ob
jective criterion was the maximization of the present value of
expected net production. A comparison of the results of the
production run to those of the net returns showed that for lac
tations 1 and 2 replacement was always earlier for the produc
tion problem where there were differences in the optimal re
placement policy, and always later than in the net returns
problem for lactations 3, 4, and 5. The differences between the
optimal production policy and the policy for the "base" net re
turns were generally greater than those among the net returns
runs.
A question of interest is if the extra cost of gathering the
data and obtaining solutions to the net returns problems can be
justified on the basis of the superiority of the decisions which
result. This is no simple question and cannot be answered def
initively on the basis of the findings of this study. If the specifi
cations of cost and revenue relationships are wellfounded, how
ever, the economist should find runs based on the net returns
criterion of greater interest and applicability than those based
on the production criterion.
Operational Employment of the Model and Costs of Computation
The optimal decision for any cow for which the relation
ships, parameters, and variables of a given replacement model
are pertinent, whether it is from the same basic herd or not,
is provided in the DP results. The potential costs to a dairyman
for the evaluation of an individual cow is largely a function of
the number of cows over which the fixed costs of solving a par
ticular DP model can be spread. This number may be limited
to an individual herd (or even some subherd group), or to some
group of herds which have certain important characteristics in
common. These characteristics in common may be breed, geo
graphical location, average production, management system, etc.
It required the IBM 360/50 about 14 minutes to solve the
net returns version of the DP model for 15 stages. Assuming
one evaluation for each cow each year, and a commercial com
puter rate of $260 per hour, the cost per cow if 100 cows are
evaluated is about 60 cents. If the model is good for 1,000 cows,
the per cow cost is about 6 cents. To these computer costs must
be added charges for processing the cow data, revising the re
placement models, mailing the results of the evaluation to the
dairymen, etc. Only actual experience in using such a model can
permit a very accurate estimate of the cost of these supporting
services.
No attempt was made to actually use the replacement model
presented in this study in an operational setting. It seems that
the most promising organizations for these purposes are the
DHIA data processing centers at either the state or regional
levels. Other possibilities exist of course, such as commercial
farm management or farm recordkeeping service organiza
tions, and the several state or regional farm recordsprocessing
centers under the sponsorship of certain LandGrant Colleges.
DHIA records processing centers already have both a store
and flow of dairy cow data, however, which with but little
modification would provide the data necessary to employ quanti
tative replacement decision models. Involved in the DHIA pro
grams, too, are the people who generally know most about the
total environment within which dairy cow replacement decisions
must be made. These people are in a good position to take the
formal findings of a decision model and put them into the per
spective which differences among herds and the objectives of
dairymen might require.
SUMMARY
The objective of the present study was to develop a forma
decision model which could use recurrent farm level data. for
the periodic evaluation and optimal replacement of dairy cows.
The production process in the dairy cow replacement problenr
was defined as any cow in a herd, all her replacements, and all
her replacements' replacements. A lactation was referred to as
a stage, and the state of the process referred to the character!
istics of whatever cow happened to occupy the process at any
given stage. Transformations of the process from one state to
another during any stage were determined by the decision that
was made regarding whether to Replace or Keep whatever cow
was in the process at any particular time. Interest centered on
considerations relevant within a replacement chain, as the del
termination of the optimal number of chains was considered to
be outside the scope of the investigation.
Two distinct phases in the development of quantitative cow
replacement models were seen. Phase I was concerned largely
with predicting the production of 305day, 4% FCM for cowd
of appropriate combinations of relevant characteristics. A
critique of the variables most likely to be related to milk pro
duction, based largely on a review of the literature, was pre
sented.
A generalized production prediction model from which para
meters could be estimated for each selected combination was
formulated. This was done through leastsquares general regres
sion analysis, with the independent variables lactation number
and length of previous calving interval incorporated as discrete
0, 1 shifters, and production in each of the two immediately
prior lactations entered in continuous form. The regression co
efficients for production in the two prior lactations, when ap
plied to production for an actual cow, constituted the variable
part of expected production for that cow in the next lactation.
To this was added a fixed part which was the sum of the regres!
sion constant and the appropriate shifters for calving interval
and lactation number. Thus, a recursive relationship was es
tablished which gave an estimate of production for cows which
were kept and did not fail. If cows were not kept (that is, were
replaced), the production of the replacement cow was set equal
to the average production for all lactations of cows in the basic
herd at the first stage, incremented by 1% at each succeeding
stage to reflect the increasing genetic capacity of the general
cow population to produce milk over time.
The economic components of the replacement problem were
limited to those costs and returns that varied as the character
istics of the individual cow varied. Value of milk produced was
seen as the most important variable return, and costs of feed
the most important item of cost. Value of calves born was the
only other item of variable return, and depended on the produc
tion of the dam and the price of veal. Depreciation in the worth
of the cow and interest on her market value were the two other
variable costs that entered into the computation of net returns.
These two latter costs depended on the lactation number that
the cow was in and on the prices and interest rates appropriate
to the particular replacement situation.
Efforts related to phase II of the study culminated in the
development and evaluation of a replacement decision model
employing the backward, recursive approach commonly referred
to in the literature as dynamic programming. It was formulated
as a decisiondetermined stochastic Markov process with re
wards, and employed the relationships and parameters that were
developed in phase I of the investigation. There were six pos
sible values for lactation and three for previous calving inter
val. Production levels in the two prior lactations were divided
into 29 discrete intervals of 250 pounds each over the range
5,000 to 12,000 pounds of milk. This yielded 6 X 3 X 29 X 29 =
15,138 possible states of the process at each stage. The tran
sitions from production interval to production interval were
deterministic, and the lactationcalving interval transitions sto
chastic.
Five 15stage analytical numerical runs of the DP model
were made using the valueiterative solution routine. Four of
these runs had as their objective criterion the maximization of
the present value of expected net returns, the other the maxi
mization of the present value of expected production.
The prices used for the first net returns run were considered
to be at the "base" or representative levels in the study. A 20%
reduction in feed price brought about a 27% increase in net
returns. A decrease in milk price of 20 % resulted in a decrease(
in net returns of 50%. A 20% increase in beef and veal priced
caused only slight reductions in net returns.
A comparison among the optimal replacement policies for
the net returns runs showed that they were relatively insensii
tive to changes in feed prices. That is, the optimal decision
changed from Keep to Replace, or vice versa, for only a few
marginal states of the process. In most instances, however, a
feed prices declined, the optimal policy called for replacement
at somewhat higher production levels. A minor effect on the
optimal policy was also evidenced for changes in the price of
milk. The direction of change was mixed, however, as in some
cases replacement was earlier, in others it was later. The max.
imum changes in policy occurred with a change in beef and veal
prices, although the changes in total net returns were smallest
The general conclusion was drawn that even moderate
changes in the relative price levels of important input or out
put factors did not alter the optimal replacement policy to an
economically significant degree when compared to the "base'
price situation. Thus, once a solution to the replacement prob
lem is found for some representative set of factor prices, i
probably would not be rewarding to solve it again simply be
cause one or more prices have changed, even if by fairly large
amounts.
One of the most serious weaknesses of this and other dairy
cow replacement studies is seen to be in the prediction of pro
duction by individual cows. In future work particular attention
might profitably be devoted to the identification, quantification,
and specification of variables which would improve the reliability
of such predictions. The payoff for the incorporation of som
measure of cow size, methods for taking account of differin
lengths of calving intervals, and the optimal time to replace
within a lactation are also in need of further study.
It seems entirely likely, based on the experiences and findings
of the present study, that the dynamic programming dairy cow
replacement decision model could be solved for a set or sets of
input parameters and variables which would have rather wide
applicability. The computational costs of finding the optimal
replacement polices would then be only very nominal. The po
tential returns, though as yet unmeasured, would be expected
to exceed those costs'by a great deal.
LITERATURE CITED
Arave, C. W. 1962. A study of genetic changes in twelve California dairy
herds. Ph.D. thesis, Department of Genetics, University of California,
Davis. University Microfilms, Ann Arbor, Michigan.
Arnold, P. T. D., R. B. Becker, and A. H. Spurlock. 1958. Dairy cattle and
their care. Bulletin No. 599, Florida Agricultural Experiment Station,
Gainesville, Florida.
Asdell, S. A. 1951. Variations in amount of culling from D.H.I.A. herds.
Journal of Dairy Science 34: 529535.
Bellman, R. 1957. Dynamic Programming. Princeton University Press,
Princeton, New Jersey.
Berry, J. C. 1945. Repeatability of averages of different numbers of lacta
tion records for comparing dairy cows. Journal of Dairy Science 28:
355366.
Boyd, L. J. 1967. Does production affect breeding efficiency? Hoard's
Dairyman, November 10, 1967. W. D. Hoard and Sons Company, Fort
Atkinson, Wisconsin.
Burt, O. R. 1965. Optimal replacement under risk. Journal of Farm Eco
nomics 47 (2): 324346.
Burt, O. R., and J. R. Allison. 1963. Farm management decisions with
dynamic programming. Journal of Farm Economics 45 (1): 121136.
Castle, 0. M., and S. R. Searle. 1957. Repeatability of dairy cow butter
fat records in New Zealand. Journal of Dairy Science 40: 12771283.
Chisholm, A. H. 1966. Criteria for determining the optimum replacement
pattern. Journal of Farm Economics 48 (1): 107112.
d'Epenoux, F. 1963. A probabilistic production and inventory problem.
Management Science 10: 98108.
Dreyfus, S. E. 1965. Dynamic Programming and the Calculus of Varia
tions. Academic Press, New York.
Eastwood, B. R., D. E. Voelker, and A. E. Freeman. 1966. Using estimated
producing ability for ranking cows within Dairy Herd Improvement
Association herds. Journal of Dairy Science 49: 748.
Empire Livestock Marketing Cooperative. 1967. Slaughter livestock and
dairy replacement price quotations, January through June, New York.
Ezekiel, M., and K. A. Fox. 1959. Methods of Correlation and Regression
Analysis. John Wiley and Sons, Inc., New York.
Falconer, D. S. 1960. Introduction to Quantitative Genetics. The Ronald
Press Company, New York.
Faris, J. E. 1960. Analytical techniques used in determining the optimum
replacement pattern. Journal of Farm Economics 42 (4): 755766.
Faustmann, M. 1849. Berechnung des werthes, welchen waldboden sowie
noch nicht haubare holzbestande fur die waldwirtschaft besitzen (De
termination of the value of forest land and forest crops for management
purposes), Allg. Forst und JagdZeitung (General Forestry and Wild
life Journal) 25: 441455.
Gaffney, M. M. 1957. Concepts of financial maturity of timber and other
assets. A. E. Information Series No. 62, Department of Agricultural
Economics, North Carolina State College, Raleigh, North Carolina.
Giaever, H. B. 1965. Optimal dairy cow replacement policies. Ph.D. thesis,
Department of Agricultural Economics, University of California, Berke
ley. University Microfilms, Ann Arbor, Michigan.
Hadley, G. 1964. Nonlinear and Dynamic Programming. AddisonWesley
Publishing Company, Reading, Massachusetts.
Halter, A. N., and W. C. White. 1962. A replacement decision process: An
application to a caged layer enterprise. Bulletin No. 677, Kentucky
Agricultural Experiment Station, Lexington, Kentucky.
Heady, E. O., N. L. Jacobsen, J. P. Madsen, and A. E. Freeman. 1964. Milk
production functions in relation to feed inputs, cow characteristics, and
environmental conditions. Research Bulletin No. 529, Iowa Agricultural
and Home Economics Experiment Station, Ames, Iowa.
Howard, R. A. 1960. Dynamic Programming and Markov Processes. M.I.T.
Press, Cambridge, Massachusetts.
Howard, R. A. 1966. Dynamic programming. Management Science 12 (5):
317348.
Hutton, R. F. 1966. A simulation technique for making management
decisions, in dairy farmingnarrative, flow chart, and computer code
description, Agricultural Economic Report No. 87, Economic Research
Service, U. S. Department of Agriculture, Washington, D. C.
Jenkins, K. B., and A. N. Halter. 1963. A multistage stochastic replace
ment decision model: Application to replacement of dairy cows. Tech
nical Bulletin No. 67, Oregon Agricultural Experiment Station, Corvallis,
Oregon.
Johansson, I. 1961. Genetic Aspects of Dairy Cattle Breeding. University
of Illinois Press, Urbana, Illinois.
Johansson, I., and A. Hansson. 1940. Causes of variation in milk and
butterfat yield of dairy cows. Kungl. Lantbr. Akad. Tidskr. 61/2: 1127.
Kemeny, J. G., and J. L. Snell. 1960. Finite Markov Processes. D. Van
Nostrand Company, Princeton, New Jersey.
Kendrick, J. F. 1953. Standardizing D.H.I.A. records in proving dairy sires.
Information Bulletin No. 162, Bureau of Dairy Industry, U. S. Depart
ment of Agriculture, Washington, D. C.
Kislev, Y., and A. Amiad. 1968. Linear and dynamic programming in
Markov chains. American Journal of Agricultural Economics 50: 111
129.
Knight, F. H. 1921. Risk, Uncertainty and Profit. HoughtonMifflin Com
pany, Boston, Massachusetts.
Lennon, H. D., and J. P. Mixner. 1958. Relation of lactation milk produc
tion in dairy cows to maximum initial milk yield and persistency of lac
tation. Journal of Dairy 'Science 41: 969976.
Lush, J. L. 1937. Animal Breeding Plans. Collegiate Press, Inc., Ames,
Iowa.
Lush, J. L. 1960. Improving dairy cattle by breeding I: Current status
and outlook. Journal of Dairy Science 43: 702706.
Mason, I. L., A. Robertson, and B. Gjelsted. 1957. The genetic connection
between body size, milk production, and efficiency in dairy cattle. Journal
of Dairy Research 24: 135143.
Morrisson, F. B., and S. H. Morrisson. 1958. Feeds and Feeding. Morrisson
Publishing Company, Ithaca, New York.
Nemhauser, G. L. 1966. Introduction to Dynamic Programming. John
Wiley and Sons, Inc., New York.
O'Bleness, G. V., and L. D. Van Vleck. 1962. Reasons for disposals of dairy
cows from New York herds. Journal of Dairy Science 45: 10871093.
Preinreich, G. A. D. 1940. The economic life of industrial equipment.
Econometrica 8: 1244.
Salisbury, G. W., and N. L. Vandemark. 1961. Physiology of Reproduction
and Artificial Insemination of Cattle. W. H. Freeman and Company,
San Francisco, California.
Smith, B. J. 1968. The Dairy Cow Replacement Problem and the Deter
mination of Optimal Replacement Policies. Ph.D. thesis, Department
of Economics, North Carolina State University, Raleigh, North Caro
lina. University Microfilms, Ann Arbor, Michigan.
Snedecor, G. W. 1956. Statistical Methods. Iowa State College Press, Ames,
Iowa.
Tabler, K. A., and R. W. Touchberry. 1955. Selection indices based on milk
and fat yield, fat percent, and type classification. Journal of Dairy
Science 38: 11551163.
Texas Agricultural Extension Service. Undated. Putting dairy herd records
to work, Publication No. D421, Agricultural and Mechanical College of
Texas, College Station, Texas.
U. S. Department of Agriculture, 1967a. Dairyherdimprovement letter,
ARS44194, Vol. 43, No. 4, Agricultural Research Service, Washington,
D. C.
U. S. Department of Agriculture. 1967b. Dairy situation. DS315, Eco
nomic Research Service, Washington, D. C.
White, W. C. 1959. The determination of an optimal replacement policy
for a continually operating egg production enterprise. Journal of Farm
Economics 41 (5): 15351542.
Winder, J. W. L., and G. I. Trant. 1961. Comments on determining the
optimal replacement pattern. Journal of Farm Economics 43 (4): 939
951.
Wing, J. M. 1960. Raising dairy herd replacements. Circular No. S118,
Florida Agricultural Experiment Stations, Gainesville, Florida.
Wing, J. M. 1963. Dairy Cattle Management Principles and Applications.
Reinhold Publishing Corporation, New York.
ACKNOWLEDGMENTS
Dr. Richard A. King, of the Department of Economics at North Carolina
State University, provided reviews and suggestions throughout the original
study on which this report is based that were always most helpful and
encouraging.
Dr. Charles J. Wilcox supplied the data from the University of Florida
experimental dairy herd used for the prediction of milk production by
individual cows. His learned counsel concerning the biological aspects of
dairy cow performance was also of great value throughout the entire
course of this investigation.
The University of Florida provided unusual aid in the form of financial
support of the project under which this investigation was carried out. This
support included considerable typing, and computer programming and com
putational time. The great bulk of the responsibility for the typing was
most capably borne by Mrs. Karen Elwood, secretary to the author through
out the preparation of the original manuscript.
HISTORIC NOTE
The publications in this collection do
not reflect current scientific knowledge
or recommendations. These texts
represent the historic publishing
record of the Institute for Food and
Agricultural Sciences and should be
used only to trace the historic work of
the Institute and its staff. Current IFAS
research may be found on the
Electronic Data Information Source
(EDIS)
site maintained by the Florida
Cooperative Extension Service.
Copyright 2005, Board of Trustees, University
of Florida
