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Selecting Groups of Sires by Computer to Maximize Herd Breeding Goals
Selecting Groups of Sires by Computer to Maximize Herd Breeding Goals M. L. M c G I L L I A R D and J. S. C L A Y Department of Dairy Science Virginia Polytechnic Institute and State University Blacksburg 24061 ABSTRACT
A computerized linear program, MAXBULL, selects sires for dairymen to obtain maximum Predicted Difference milk while maintaining other goals. Goals are minimum average Predicted Differences fat percent and type, six individual type traits, and maximum average semen price. Individual traits for bulls are coded +1, 0 , - 1 according to evaluations from breed associations. Dairymen restrict bulls offered to the program by specifying unacceptable repeatability, Predicted Differences, and price. They further control breed, stud organizations, minimum bull numbers, use of sires with low repeatability (<50%) and without type proofs, and maximum use of specific bulls and sons of sires. Goals for individual type are specified as the percentage of cows in the herd needing improvement in each trait. The solution contains at least that percentage of semen from sires coded +1 in each trait. The dairyman receives a count of bulls surviving each of his edits, a reiteration of his goals, a list of recommended units and all information for each sire, averages for all traits weighted by the units recommended, and the expected increase in Predicted Difference milk from relaxing each goal one unit. MAXBULL has led to individualized breeding programs characterized by reasonable goals, balance in emphasis among traits, and stability in goals over time. INTRODUCTION
Dairymen can choose from more than 1,000 dairy sires in artificial insemination (AI) to breed their cows, with most sires evaluated for several production and type traits. Because correlations among many of these traits are
Received June 30, 1982. 1983 J Dairy Sci 66:647-653
negative and price per unit of semen varies extensively among bulls, it is not obvious to an individual dairyman which bulls satisfy his goals at a reasonable price. Extension dairymen in several states have promoted selection of sires that are affordable and highest for Predicted Difference (PD) dollars. Other extension dairymen have encouraged establishment of goals for the herd for several traits and use of sires that satisfy the goals with the highest PD dollar average. Sires exceeding goals for several traits are usually few and expensive. A combination of sires, meeting the goals as a group, is a desirable alternative but nearly impossible to assemble without computer. McGilliard (4) formulated a linear programming procedure to choose AI bulls and assess relative influences of PD milk, PD fat %, and PD type on minimum semen expense. Shanks and Freeman (8) used linear programming in a portfolio analysis to select from a group of sires. They also minimized semen p r i c e but coded type traits into categories. More recently, Helgren and Shook ( 1 ) h a v e been investigating nonlinear programming to include a method of averting risk. OBJECTIVE
Our purpose was to develop a practical linear program (MAXBULL) to maximize PD milk while selecting sires to fulfill goals designated by dairymen. The PD milk is maximized rather than price minimized because of an overwhelming preference by dairymen (5). Most of them find it easier to specify price as a contraint because they have budgeted breeding expenses for several years. Maximization of PD milk also provides automatic improvement as genetic quality of sires increases over time. The system was designed to solicit from dairymen relatively simple goals and specifications and submit them to a computer storing a file containing trait evaluations of each AI bull. The computer then uses a linear program 647
648
McGILLIARD AND CLAY
to select a group of bulls to obtain the goals. A list of these bulls is returned to the d a i r y m a n with c o m p u t e r i z e d s u m m a r y and evaluation o f recommendations. EDITS A N D T R A I T S
Several traits are used to edit bulls. Bulls surviving all edits are offered to the linear program. Edit traits include breed, m i n i m u m repeatability, PD milk, PD fat %, PD dollars, PD type, m a x i m u m price per unit, and a choice of 10 organizations. Up to 1 0 b u l l s also can be p r e e m p t e d arbitrarily by code number. Variables required for each bull are in Table 1. Bulls n o t evaluated for PD type are either excluded or coded 0.0 or - 1 . 0 by choice of the dairyman with individual t y p e traits coded 0. Individual t y p e i n f o r m a t i o n is n o t used in the edits but is c o d e d for rear udder, fore udder, udder support, teat size and placement, hind legs, and feet. Holstein individual t y p e is f r o m the Holstein Association (2) and is coded +1 for a trait if the bull has sired a significant n u m b e r of daughters correct in the trait, - 1 if his daughters are deficient, and 0 if neither, both, or n o t evaluated. Individual t y p e for o t h e r breeds is f r o m their respective breed organizations and is coded 1, - 1 , 0 f r o m their c o n t i n u o u s scales such that one-third of the available AI bulls are in each code. This coding system facilitates a simple definition of the individual t y p e goals to be provided by the dairyman.
TABLE 1. Variables required for each bull. Breed Stud Code number Sire code number Repeatability Predicted Difference (PD) milk PD Fat % PD Fat PD Dollars
Repeatability type PD Type Udder support code Rear udder code Fore udder code Teat code Hind leg code Feet code Price
use of bulls with less than 50% repeatability (Low-R buUs) is limited to a percentage designated by the dairyman. Finally, up to six sires of sons can be restricted in percentage o f units o f semen r e c o m m e n d e d f r o m sons of each sire. This feature is e x t r e m e l y useful for reducing the potential for inbreeding when some sires have m a n y sons, as r e p o r t e d by R u t t e r and Pearson (7). The i n p u t f o r m used by d a i r y m e n is in Figure 1. This f o r m u l a t i o n yields a linear p r o g r a m m i n g tableau o f m rows and n columns where m = (10) constraints + (1) unit equality + (1) objective r o w + n u m b e r o f sires to constrain sons n = ( m - 2 ) slack variables + n u m b e r of bulls offered to the program.
TABLE 2. Variables constrained by the linear program. L I N E A R MODEL F O R M U L A T I O N
The linear program selects bulls and n u m b e r of units of semen f r o m each to m a x i m i z e the weighted average PD milk of the solution. The objective f u n c t i o n maximizes k
k NiPDMi/~ N i i=l i=l where k is the n u m b e r o f bulls offered to the program, N i is the n u m b e r of units o f bull i, and PDM i is the PD milk of bull i. Constraints are summarized in Table 2. M i n i m u m weighted averages are specified as goals by the dairyman for PD fat %, PD type, udder support, rear udder, fore udder, teats, hind legs, and feet. A m a x i m u m weighted average is specified for price. To reduce risk, Journal of Dairy Science Vol. 66, No. 3, 1983
Variable
Constraint
Number of units Predicted Difference (PD) fat PD type Udder support Rear udder Fore udder Teats Hind legs Feet Price Low-R bulls a Sons of a sire Units from a bullb Units from a low-R bull b
Exact total Minimum average Minimum average Minimum average Minimum average Minimum average Minimum average Minimum average Minimum average Maximum average Maximum percentage Maximum percentage Maximum percentage Maximum percentage
aBulls with repeatability less than 50% for PD milk. bNot constrained until cycle 2.
OUR INDUSTRY TODAY
649
MAXBULL Computerized Sire Selection The computer will use your preferences to choose bulls to maximize the Predicted Difference Milk in your breeding program.
BULL RESTRICTIONS
GOALS OF YOUR BREEDING PROGRAM These are the m i n i m u m averases you wish to obtain with the bulls used
Place these limits on individual bulls. (Be sure these are less than your goal8 on the opposite side.)
Averase PD Fat %
There will be no bulls used with:
=
or higher
Average PD Type
or higher
Less than
Repeatability
Average Price/Unit =
or lower
Less than
PD Dollars
(Direct herd w/o technician) % of cows needing improvement in: US
Udder Support
Check here for no emphasis:
or %
US
RU
Rear Udder
%
RU
FU
Fore Udder
%
FU
TT
Teat Size & PI
%
TT
HL
Hind Legs
%
HL
FT
Feet
%
Less than
PD Milk
Less than
PD Fat%
Less than
PD Type
More than
Prlce/Unit
I f you wish to l i m i t semen, w r i t e code nos. and maximum % o f t o t a l u n i t s . L i m i t sons o f L i m i t these b u l l s : % these s i r e s : % %
%
%
%
%
%
FT
NUMBER OF BULLS
BREED AND STUD RESTRICTIONS
(A Low-R bull is less than 50% Repeatability
Check your breed:
Holstein
No. of units you wish to purchase (6 mos, optional)
Brown Swiss (5) _ _
Jersey
Ayrshire
Guernsey
Units
M i n i m u m number of A.I. bulls to use
Bulls
Maximum % of cows to be bred to Low-R bulls
%
M a x i m u m % of cows to be bred to any on___eeLow-R bull
%
(i)
Select
(7)
ABS (29) Curtiss
Exclude them Use 0.0 Use -i.0
(2)
Check the studs you are willing to use:
(40)
Carnation What do you want done with bulls having no PDT?
(3)
(4)
M i n n .
(3) (21)
Atlantic
(11,24,23)
Valley
Eastern Midwest
(17)
(15)
Sire Power _ _
Tri-State
(9) (14
Name: Address: Return to:
Dr. Bennet C. Cassell Dairy Science, Virgi,ia Tech Blacksburg, VA 24061
Phone: County:
Figure 1. MAXBULL input form used by dairymen.
For 4 0 0 bulls and no son constraints, m × n would be 12 × 4 1 0 and for 50 bulls, 12 x 60. Ninety percent of the solutions to this formulation (cycle 1) are unacceptable because
of too f e w bulls r e c o m m e n d e d or too m a n y units from particular bulls. Criteria are in the next section and Table 3. These conditions are checked, and a n e w linear program is formulated Journal of Dairy Science Vol. 66, No. 3, 1983
6 50
McGILLIARD AND CLAY
(cycle 2). The optimal properties of cycle 1 often are set aside in favor of an additional formulation (cycle 2) that conforms to dairymen's expectations of balance of use among bulls recommended.
MAXBULLFLOWCHART Enter Restrictions
I I Cycle 2 ~ Fill LP Matrixa I Enter Goals (Cycle l) I Solve LP
Enter and Edit Bulls
CONTROLLING NUMBER OF BULLS AND UNITS
The 50 bulls from cycle 1 that were closest to entering the solution are included in formulation for cycle 2. They are constrained by percentage use and continue to serve as activities as in cycle 1. These constraints are necessary to assure a minimum number of bulls in the solution as well as an acceptable distribution of units among bulls. Table 3 lists, for each number of bulls requested, the maximum limit on units from the first bull from cycle 1 and the minimum number of units for a bull to count toward the number of bulls requested by the dairyman. The most recommended bull from cycle 1 is limited in units to twice the units of the second most recommended but not more than the units in Table 3. The second bull from cycle 1 subsequently is limited to his units in cycle 1, and the m a x i m u m units for each remaining sire are determined by dividing by ( n - 2 ) the units remaining if the first and second bulls are used to capacity. For example, if the first 2 bulls from cycle 1 were 50 and 30% of the units requested where at least 5 bulls were desired, then in cycle 2 the second bull would be limited to 30% and the first bull to 33% from Table 3, because it is less than twice 30%. The remaining 48 bulls would each be limited to ( 1 0 0 - 3 3 - 3 0 ) % / ( 5 - 2 ) or 12%.
TABLE 3. Restrictions on units from individual bulls. No. bulls requested
Maximum a %
Minimum b % to count
1 2 3 4 5 6 10
100 67 50 40 33 30 18
50 25 17 12 10 8 5
aMaximum = 200/(no. bulls + 1). bMinimum = lO0/(2(no, bulls)). Journal of Dairy Science Vol. 66, No. 3, 1983
/
Solution a Cycle l /
k
/ \ ~
No. Bulls I Excess Units I Indiv. Bulls ~ Fail[
No Solution or Cycle 2
\ \
\
R~ p r ~ ~
-*-Assign Units to Individual Bulls aLinear programming tableau. Figure 2. Flowchart of MAXBULL selection program.
That would assure at least 5 bulls in the solution. Should the 12% be less than the minimum to count as a bull, additional reductions would be made in units allowed the second bull. Two additional constraints can be applied to the 50 bulls. Each low-R bull is constrained if necessary to the percentage of units specified by the dairyman for individual Low-R bulls. Also, up to 10 bulls may be restricted arbitrarily in percentage by code number from the dairyman or operator. The linear programming tableau for cycle 2 would be m x n, where m = 12 + number of sires to constrain sons + number of bulls offered (<50) n = ( m - 2 ) slack variables + number of bulls offered (~<50) Maximum m × n dimensions for cycle 2 are 68 x 116. A flowchart of this procedure is in
OUR INDUSTRY TODAY ******************************************************
651
******************************************************
RAXBULL
SAMPLE UAIRYMAN MINIMUM # BULLS=
5.
MAXIMUM~ LOW R BULLS= 25.
MAXIMUM% PER LOW R BULL= 15.
BULL EDITING MISS. BREED R PDMILK PDFAT% PDTYPE PDS PRICE PDIYPE STUD(S) .............................................................................................
EDIT L I M I T S
3
35
lOUD
-
.30
-1.OO
150
3O.OO
O
SURVIVORS 411 409 354 352 336 218 202 .......................................................................................................... BULLS L I M I T E D ( ~ ) 10. 7 H l 1 1 0 BULL SIRE LIMITS ( ~ )
A- 7H58
50.;
O
3
7
9 11 1~ 15 17 21 2~ 29 N0
202
202
B- NOHRO25 1DO.;
SUGGESIED BREEDING PROGRAM FOR MAXIMUM PDMILK ..................... =. . . . . . . . . . . . . . . . . . . . . . . DESCRIPTIVE TYPE - % ONES ........................... UNITS R PDMILR CONF PDFAT~ PDFAT PDS R-T PDTYPE US RU FU TT ................................................................................................. GOALS }50 O MAX - .08 1.00 30 15 20 -100 ................................................................................................. BULL BS ................................... 7H6O3 . . . .
7H556 3H922
29H3098 7H1120 3H672
LANCE TROY DYNAMO ASTRO CHIEF CAVALIER
A
NO.
25
65
,BB~
2,.
37 30 20 ~O 5
25 20 13 13 3
75 65 71 37 98
19h7 1888 1874 1,39 2027
232 274
-
OB
.23 .08 ,lO
249
-
368 65
O - O .19
=.........................................
BONUSES (AVERAGE MINUS GOAL)
65
FEBRUARY 1982 IIERAIION NUMBER= 55 OPT=O; CYCLE NUMBER: I
NEW FEATURE -
PRICE
O
12.OO
55
226
58
T3
0
O
O
i
,
0
.....
33 81 52 52 "3
~92 271 220 189 215
62 ~ T 33 95
1.63 0 1.51 1.,1 .75
0 ~
1 ~
1 ?
0 ~
I ?
0 ~
1 1
1 1
1
1
O 1
I0.00 15.00 18.00 9.00 12.OO
h7
1.DO
30
55
15
53
3
12.00
115
h3
3
0.0
SONS OF ANY SIPE CAN BE LIMITED.
. . 1 55
.
.
==== . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
MILK CHANGE FROM RELAXING O EACH GOAL ONE UNIT .........................................................
****
FT
IO
%
3R
............................................................................................................................ WEIGHTED AVERAGES 150 65 1844 265 - .08 53 220 .................
HL
0 IB /.01%
O 230
39 O
1
======= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BENNET G. CASSELL EXTENSION GENETICISI
34 O
O
O
O
31
=== . . . . . . . . . . . . . . . . . . .
MIRE MCGILLIARD & JOHN CLAY DAIRY SCIENCE - VIRGINIA TECH BLACKSBURG 2ND61
AFTER BULL NAME: A:ELEVATION SON, B=CHIEF SON.
Figure 3. MAXBULL printout for dairyman.
Figure 2. On an international Business Machines 3032 computer, MAXBULL requires 350K m e m o r y and about 3 s central processing unit time per dairyman. BREEDING PROGRAMS
The dairyman receives the computer printout depicted in Figure 3. It reiterates his requests, shows him the number of bulls surviving his edits, and lists all traits and averages for the group of bulls recommended to meet his goals and maximize average PD milk. For individual type, the dairyman is provided enough units from bulls coded +1 to equal or surpass the percentage of his herd needing improvement in each trait. This is measured as percentage of net ones, equal to an average of the bulls' descriptive type codes for a particular trait and weighted by each bull's usage. A zero for net ones would indicate as many units coded +1 as coded - 1 , not necessarily regression in the trait. Percentage net ones ranges from - 1 0 0 to 100.
Bonuses indicate which weighted averages exceed the goals. For traits meeting goals exactly, the milk change quantifies the PD milk to be gained by relaxing the goal one unit, or .01% for PD fat %. These marginal changes are not linear, however, and cannot be extrapolated accurately. If an additional do1Iar is expected to yield at least $2.50 in present value of milk at the requested PD fat %, the dairyman is advised beneath the milk change line to increase his price constraint. If the present value of milk is less than the average price of semen, he is advised that production alone is not expected to return his investment in 10 yr at 10% interest. These formulae are adapted from (3). Finally, if no solution is obtained, the most binding constraint is listed. Usually the operator adjusts immediately the edits or goals to obtain a solution without conferring again with the dairyman. It is critical that the operator understand the function of the program relative to dairymen's desires. Journal of Dairy Science Vol. 66, No. 3, 1983
65 2
McGILLIARD AND CLAY RESULTS
Goals from dairymen have by design come only through educational meetings and residual requests. Several unsolicited requests have been received from across the country. The vast majority of MAXBULL programs have been for Holstein dairymen, although several Jersey and a few Brown Swiss, Ayrshire, and Guernsey breeders have obtained programs. Table 4 lists average goals and solutions for 196 Holstein dairymen whose requests were processed in February, 1982. These 196 include only dairymen and not individuals representing industry and education who also have tried the program. These averages relative to individual sires in the MAXBULL file are equivalent to the 90th percentile for PD milk, 60th for PD fat %, 90th for PD dollars, and 65th for PD type. Milk and fat % are negatively correlated among breeding programs, and udder support is generally the most restrictive descriptive trait although not least available among the sires. A more complete analysis of these data has been made by McGilliard and Clay (6). PROBLEMS
Maintenance of sire information is tedious, because prices must be obtained from studs, and type information from breed organizations. The Holstein Association does not distribute type information on tape to extension dairymen or dairy records processing centers.
Excessively restrictive editing and impractical goals from dairymen cause delays in returning usable solutions. On occasion solutions will contain too many bulls, several of which may represent only single units. This occurs most frequently with requests for bulls from only one stud. The problem is alleviated by modifying the request for fewer bulls or m i n i m u m units per bull. Repeatabilities are not high and, except for elimination of Low-R bulls, can only be controlled through edits. No dairyman has objected; some individuals consider this an asset relative to genetic trend. By request, a solution can be obtained for minimum groups of units, such as 4 or 10. This, however, sacrifices the exactness of the solution. REFINEMENTS
An interactive version of MAXBULL has been developed. It queries the user for information, produces breeding programs in less than 1 min, and allows immediate editing and storage of goals. For dairymen owning semen, an inventory program will adjust their goals such that MAXBULL will suggest a set of bulls to complement their inventory. Minimum average repeatability is a possible constraint which has been suppressed in favor of the Low-R concept. Unless repeatability is tied to another trait in the matrix, bulls inferior in all traits will be chosen only to increase repeatability. A common request is to maximize
TABLE 4. Averages of MAXBULL recommendations for 196 Holstein dairymen in February 1981. Goal Trait Predicted difference (PD) milk (kg) PD Fat (%) PD Fat (kg) Repeatability (%) PD Dollars PD Type (pts) Udder support (%) Rear udder (%) Fore udder (%) Teat size, placement (%) Hind legs (%) Feet (%) Price (S/unit)
X
Solution SD
--.05
.05
.58 23 20 21 14 22 22 19
Journal of Dairy Science Vol. 66, No. 3, 1983
.56 17 16 17 14 17 16 13
X
SD
789 --.05 25 67 215 .85 29 35 36 27 33 46 18
90 .04 3 6 22 .41 19 29 23 22 21 24 12
OUR INDUSTRY TODAY PD dollar value. In MAXBULL, PD dollar value is maximized at the PD fat % specified by the dairyman. Several additional features may be plausible for future MAXBULL versions. Possibilities are calving difficulty information, linear individual type including additional traits such as stature, a maximum average genetic relationship among bulls in a solution, and percentage limits on units from various studs. An option could allow maximization of variables other than PD milk, including an individualized selection index or, to minimize risk, the lower limit of a 60% confidence interval on PD milk. CONCLUSIONS
Using the MAXBULL procedure has produced objective breeding programs tailored to the goals of individual dairymen. The solutions obtain maximum PD milk superior to normal programs while maintaining balance among the other traits. Storing requests from dairymen stabilizes their goals over time and yields automatic improvement in PD xlollar averages as the quality of the bull population progresses.
653 ACKNOWLEDGMENTS
B. R. Bell, J. E. Honnette, and N. T. Jones contributed to the initial concept of selecting sires by linear programming, and J. C. Kelley developed the interactive components. REFERENCES
1 Helgren, R. E., G. E. Shook, and R. A. Schoney. 1980. A portfolio model for dairy sire selection. J. Dairy Sci. 63(Suppl. 1): 109. 2 Holstein Association. 1982. Sire summaries 1982. Volume 1. Holstein-Friesian Assoc. Am., Brattleboro, VT. 3 McGilliard, M. L. 1978. Guidelines for purchasing semen. J. Dairy Sci. 61:1680. 4 McGilliard, M. L. 1979. Relationships among genetic goals and herd breeding expenses. J. Dairy Sci. 62:85. 5 McGilliard, M. L. 1980. Informal opinions of dairymen in Virginia. 6 McGilliard, M. L., and J. S. Clay. 1983. Breeding programs of dairymen selecting Holstein sires by computer. J. Dairy Sci. 66:654. 7 Rutter, C. R., and R. E. Pearson. 1981. Breeding structure in artificial insemination Holsteins. J. Dairy Sci. 64(Suppl. 1):74. 8 Shanks, R. D., and A. E. Freeman. 1979. Choosing progeny-tested Holstein sires that meet genetic goals at minimum cost. J. Dairy Sci. 62:1429.
Journal of Dairy Science Vol. 66, No. 3, 1983
A computerized linear program, MAXBULL, selects sires for dairymen to obtain maximum Predicted Difference milk while maintaining other goals. Goals are minimum average Predicted Differences fat percent and type, six individual type traits, and maximum average semen price. Individual traits for bulls are coded +1, 0 , - 1 according to evaluations from breed associations. Dairymen restrict bulls offered to the program by specifying unacceptable repeatability, Predicted Differences, and price. They further control breed, stud organizations, minimum bull numbers, use of sires with low repeatability (<50%) and without type proofs, and maximum use of specific bulls and sons of sires. Goals for individual type are specified as the percentage of cows in the herd needing improvement in each trait. The solution contains at least that percentage of semen from sires coded +1 in each trait. The dairyman receives a count of bulls surviving each of his edits, a reiteration of his goals, a list of recommended units and all information for each sire, averages for all traits weighted by the units recommended, and the expected increase in Predicted Difference milk from relaxing each goal one unit. MAXBULL has led to individualized breeding programs characterized by reasonable goals, balance in emphasis among traits, and stability in goals over time. INTRODUCTION
Dairymen can choose from more than 1,000 dairy sires in artificial insemination (AI) to breed their cows, with most sires evaluated for several production and type traits. Because correlations among many of these traits are
Received June 30, 1982. 1983 J Dairy Sci 66:647-653
negative and price per unit of semen varies extensively among bulls, it is not obvious to an individual dairyman which bulls satisfy his goals at a reasonable price. Extension dairymen in several states have promoted selection of sires that are affordable and highest for Predicted Difference (PD) dollars. Other extension dairymen have encouraged establishment of goals for the herd for several traits and use of sires that satisfy the goals with the highest PD dollar average. Sires exceeding goals for several traits are usually few and expensive. A combination of sires, meeting the goals as a group, is a desirable alternative but nearly impossible to assemble without computer. McGilliard (4) formulated a linear programming procedure to choose AI bulls and assess relative influences of PD milk, PD fat %, and PD type on minimum semen expense. Shanks and Freeman (8) used linear programming in a portfolio analysis to select from a group of sires. They also minimized semen p r i c e but coded type traits into categories. More recently, Helgren and Shook ( 1 ) h a v e been investigating nonlinear programming to include a method of averting risk. OBJECTIVE
Our purpose was to develop a practical linear program (MAXBULL) to maximize PD milk while selecting sires to fulfill goals designated by dairymen. The PD milk is maximized rather than price minimized because of an overwhelming preference by dairymen (5). Most of them find it easier to specify price as a contraint because they have budgeted breeding expenses for several years. Maximization of PD milk also provides automatic improvement as genetic quality of sires increases over time. The system was designed to solicit from dairymen relatively simple goals and specifications and submit them to a computer storing a file containing trait evaluations of each AI bull. The computer then uses a linear program 647
648
McGILLIARD AND CLAY
to select a group of bulls to obtain the goals. A list of these bulls is returned to the d a i r y m a n with c o m p u t e r i z e d s u m m a r y and evaluation o f recommendations. EDITS A N D T R A I T S
Several traits are used to edit bulls. Bulls surviving all edits are offered to the linear program. Edit traits include breed, m i n i m u m repeatability, PD milk, PD fat %, PD dollars, PD type, m a x i m u m price per unit, and a choice of 10 organizations. Up to 1 0 b u l l s also can be p r e e m p t e d arbitrarily by code number. Variables required for each bull are in Table 1. Bulls n o t evaluated for PD type are either excluded or coded 0.0 or - 1 . 0 by choice of the dairyman with individual t y p e traits coded 0. Individual t y p e i n f o r m a t i o n is n o t used in the edits but is c o d e d for rear udder, fore udder, udder support, teat size and placement, hind legs, and feet. Holstein individual t y p e is f r o m the Holstein Association (2) and is coded +1 for a trait if the bull has sired a significant n u m b e r of daughters correct in the trait, - 1 if his daughters are deficient, and 0 if neither, both, or n o t evaluated. Individual t y p e for o t h e r breeds is f r o m their respective breed organizations and is coded 1, - 1 , 0 f r o m their c o n t i n u o u s scales such that one-third of the available AI bulls are in each code. This coding system facilitates a simple definition of the individual t y p e goals to be provided by the dairyman.
TABLE 1. Variables required for each bull. Breed Stud Code number Sire code number Repeatability Predicted Difference (PD) milk PD Fat % PD Fat PD Dollars
Repeatability type PD Type Udder support code Rear udder code Fore udder code Teat code Hind leg code Feet code Price
use of bulls with less than 50% repeatability (Low-R buUs) is limited to a percentage designated by the dairyman. Finally, up to six sires of sons can be restricted in percentage o f units o f semen r e c o m m e n d e d f r o m sons of each sire. This feature is e x t r e m e l y useful for reducing the potential for inbreeding when some sires have m a n y sons, as r e p o r t e d by R u t t e r and Pearson (7). The i n p u t f o r m used by d a i r y m e n is in Figure 1. This f o r m u l a t i o n yields a linear p r o g r a m m i n g tableau o f m rows and n columns where m = (10) constraints + (1) unit equality + (1) objective r o w + n u m b e r o f sires to constrain sons n = ( m - 2 ) slack variables + n u m b e r of bulls offered to the program.
TABLE 2. Variables constrained by the linear program. L I N E A R MODEL F O R M U L A T I O N
The linear program selects bulls and n u m b e r of units of semen f r o m each to m a x i m i z e the weighted average PD milk of the solution. The objective f u n c t i o n maximizes k
k NiPDMi/~ N i i=l i=l where k is the n u m b e r o f bulls offered to the program, N i is the n u m b e r of units o f bull i, and PDM i is the PD milk of bull i. Constraints are summarized in Table 2. M i n i m u m weighted averages are specified as goals by the dairyman for PD fat %, PD type, udder support, rear udder, fore udder, teats, hind legs, and feet. A m a x i m u m weighted average is specified for price. To reduce risk, Journal of Dairy Science Vol. 66, No. 3, 1983
Variable
Constraint
Number of units Predicted Difference (PD) fat PD type Udder support Rear udder Fore udder Teats Hind legs Feet Price Low-R bulls a Sons of a sire Units from a bullb Units from a low-R bull b
Exact total Minimum average Minimum average Minimum average Minimum average Minimum average Minimum average Minimum average Minimum average Maximum average Maximum percentage Maximum percentage Maximum percentage Maximum percentage
aBulls with repeatability less than 50% for PD milk. bNot constrained until cycle 2.
OUR INDUSTRY TODAY
649
MAXBULL Computerized Sire Selection The computer will use your preferences to choose bulls to maximize the Predicted Difference Milk in your breeding program.
BULL RESTRICTIONS
GOALS OF YOUR BREEDING PROGRAM These are the m i n i m u m averases you wish to obtain with the bulls used
Place these limits on individual bulls. (Be sure these are less than your goal8 on the opposite side.)
Averase PD Fat %
There will be no bulls used with:
=
or higher
Average PD Type
or higher
Less than
Repeatability
Average Price/Unit =
or lower
Less than
PD Dollars
(Direct herd w/o technician) % of cows needing improvement in: US
Udder Support
Check here for no emphasis:
or %
US
RU
Rear Udder
%
RU
FU
Fore Udder
%
FU
TT
Teat Size & PI
%
TT
HL
Hind Legs
%
HL
FT
Feet
%
Less than
PD Milk
Less than
PD Fat%
Less than
PD Type
More than
Prlce/Unit
I f you wish to l i m i t semen, w r i t e code nos. and maximum % o f t o t a l u n i t s . L i m i t sons o f L i m i t these b u l l s : % these s i r e s : % %
%
%
%
%
%
FT
NUMBER OF BULLS
BREED AND STUD RESTRICTIONS
(A Low-R bull is less than 50% Repeatability
Check your breed:
Holstein
No. of units you wish to purchase (6 mos, optional)
Brown Swiss (5) _ _
Jersey
Ayrshire
Guernsey
Units
M i n i m u m number of A.I. bulls to use
Bulls
Maximum % of cows to be bred to Low-R bulls
%
M a x i m u m % of cows to be bred to any on___eeLow-R bull
%
(i)
Select
(7)
ABS (29) Curtiss
Exclude them Use 0.0 Use -i.0
(2)
Check the studs you are willing to use:
(40)
Carnation What do you want done with bulls having no PDT?
(3)
(4)
M i n n .
(3) (21)
Atlantic
(11,24,23)
Valley
Eastern Midwest
(17)
(15)
Sire Power _ _
Tri-State
(9) (14
Name: Address: Return to:
Dr. Bennet C. Cassell Dairy Science, Virgi,ia Tech Blacksburg, VA 24061
Phone: County:
Figure 1. MAXBULL input form used by dairymen.
For 4 0 0 bulls and no son constraints, m × n would be 12 × 4 1 0 and for 50 bulls, 12 x 60. Ninety percent of the solutions to this formulation (cycle 1) are unacceptable because
of too f e w bulls r e c o m m e n d e d or too m a n y units from particular bulls. Criteria are in the next section and Table 3. These conditions are checked, and a n e w linear program is formulated Journal of Dairy Science Vol. 66, No. 3, 1983
6 50
McGILLIARD AND CLAY
(cycle 2). The optimal properties of cycle 1 often are set aside in favor of an additional formulation (cycle 2) that conforms to dairymen's expectations of balance of use among bulls recommended.
MAXBULLFLOWCHART Enter Restrictions
I I Cycle 2 ~ Fill LP Matrixa I Enter Goals (Cycle l) I Solve LP
Enter and Edit Bulls
CONTROLLING NUMBER OF BULLS AND UNITS
The 50 bulls from cycle 1 that were closest to entering the solution are included in formulation for cycle 2. They are constrained by percentage use and continue to serve as activities as in cycle 1. These constraints are necessary to assure a minimum number of bulls in the solution as well as an acceptable distribution of units among bulls. Table 3 lists, for each number of bulls requested, the maximum limit on units from the first bull from cycle 1 and the minimum number of units for a bull to count toward the number of bulls requested by the dairyman. The most recommended bull from cycle 1 is limited in units to twice the units of the second most recommended but not more than the units in Table 3. The second bull from cycle 1 subsequently is limited to his units in cycle 1, and the m a x i m u m units for each remaining sire are determined by dividing by ( n - 2 ) the units remaining if the first and second bulls are used to capacity. For example, if the first 2 bulls from cycle 1 were 50 and 30% of the units requested where at least 5 bulls were desired, then in cycle 2 the second bull would be limited to 30% and the first bull to 33% from Table 3, because it is less than twice 30%. The remaining 48 bulls would each be limited to ( 1 0 0 - 3 3 - 3 0 ) % / ( 5 - 2 ) or 12%.
TABLE 3. Restrictions on units from individual bulls. No. bulls requested
Maximum a %
Minimum b % to count
1 2 3 4 5 6 10
100 67 50 40 33 30 18
50 25 17 12 10 8 5
aMaximum = 200/(no. bulls + 1). bMinimum = lO0/(2(no, bulls)). Journal of Dairy Science Vol. 66, No. 3, 1983
/
Solution a Cycle l /
k
/ \ ~
No. Bulls I Excess Units I Indiv. Bulls ~ Fail[
No Solution or Cycle 2
\ \
\
R~ p r ~ ~
-*-Assign Units to Individual Bulls aLinear programming tableau. Figure 2. Flowchart of MAXBULL selection program.
That would assure at least 5 bulls in the solution. Should the 12% be less than the minimum to count as a bull, additional reductions would be made in units allowed the second bull. Two additional constraints can be applied to the 50 bulls. Each low-R bull is constrained if necessary to the percentage of units specified by the dairyman for individual Low-R bulls. Also, up to 10 bulls may be restricted arbitrarily in percentage by code number from the dairyman or operator. The linear programming tableau for cycle 2 would be m x n, where m = 12 + number of sires to constrain sons + number of bulls offered (<50) n = ( m - 2 ) slack variables + number of bulls offered (~<50) Maximum m × n dimensions for cycle 2 are 68 x 116. A flowchart of this procedure is in
OUR INDUSTRY TODAY ******************************************************
651
******************************************************
RAXBULL
SAMPLE UAIRYMAN MINIMUM # BULLS=
5.
MAXIMUM~ LOW R BULLS= 25.
MAXIMUM% PER LOW R BULL= 15.
BULL EDITING MISS. BREED R PDMILK PDFAT% PDTYPE PDS PRICE PDIYPE STUD(S) .............................................................................................
EDIT L I M I T S
3
35
lOUD
-
.30
-1.OO
150
3O.OO
O
SURVIVORS 411 409 354 352 336 218 202 .......................................................................................................... BULLS L I M I T E D ( ~ ) 10. 7 H l 1 1 0 BULL SIRE LIMITS ( ~ )
A- 7H58
50.;
O
3
7
9 11 1~ 15 17 21 2~ 29 N0
202
202
B- NOHRO25 1DO.;
SUGGESIED BREEDING PROGRAM FOR MAXIMUM PDMILK ..................... =. . . . . . . . . . . . . . . . . . . . . . . DESCRIPTIVE TYPE - % ONES ........................... UNITS R PDMILR CONF PDFAT~ PDFAT PDS R-T PDTYPE US RU FU TT ................................................................................................. GOALS }50 O MAX - .08 1.00 30 15 20 -100 ................................................................................................. BULL BS ................................... 7H6O3 . . . .
7H556 3H922
29H3098 7H1120 3H672
LANCE TROY DYNAMO ASTRO CHIEF CAVALIER
A
NO.
25
65
,BB~
2,.
37 30 20 ~O 5
25 20 13 13 3
75 65 71 37 98
19h7 1888 1874 1,39 2027
232 274
-
OB
.23 .08 ,lO
249
-
368 65
O - O .19
=.........................................
BONUSES (AVERAGE MINUS GOAL)
65
FEBRUARY 1982 IIERAIION NUMBER= 55 OPT=O; CYCLE NUMBER: I
NEW FEATURE -
PRICE
O
12.OO
55
226
58
T3
0
O
O
i
,
0
.....
33 81 52 52 "3
~92 271 220 189 215
62 ~ T 33 95
1.63 0 1.51 1.,1 .75
0 ~
1 ~
1 ?
0 ~
I ?
0 ~
1 1
1 1
1
1
O 1
I0.00 15.00 18.00 9.00 12.OO
h7
1.DO
30
55
15
53
3
12.00
115
h3
3
0.0
SONS OF ANY SIPE CAN BE LIMITED.
. . 1 55
.
.
==== . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
MILK CHANGE FROM RELAXING O EACH GOAL ONE UNIT .........................................................
****
FT
IO
%
3R
............................................................................................................................ WEIGHTED AVERAGES 150 65 1844 265 - .08 53 220 .................
HL
0 IB /.01%
O 230
39 O
1
======= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BENNET G. CASSELL EXTENSION GENETICISI
34 O
O
O
O
31
=== . . . . . . . . . . . . . . . . . . .
MIRE MCGILLIARD & JOHN CLAY DAIRY SCIENCE - VIRGINIA TECH BLACKSBURG 2ND61
AFTER BULL NAME: A:ELEVATION SON, B=CHIEF SON.
Figure 3. MAXBULL printout for dairyman.
Figure 2. On an international Business Machines 3032 computer, MAXBULL requires 350K m e m o r y and about 3 s central processing unit time per dairyman. BREEDING PROGRAMS
The dairyman receives the computer printout depicted in Figure 3. It reiterates his requests, shows him the number of bulls surviving his edits, and lists all traits and averages for the group of bulls recommended to meet his goals and maximize average PD milk. For individual type, the dairyman is provided enough units from bulls coded +1 to equal or surpass the percentage of his herd needing improvement in each trait. This is measured as percentage of net ones, equal to an average of the bulls' descriptive type codes for a particular trait and weighted by each bull's usage. A zero for net ones would indicate as many units coded +1 as coded - 1 , not necessarily regression in the trait. Percentage net ones ranges from - 1 0 0 to 100.
Bonuses indicate which weighted averages exceed the goals. For traits meeting goals exactly, the milk change quantifies the PD milk to be gained by relaxing the goal one unit, or .01% for PD fat %. These marginal changes are not linear, however, and cannot be extrapolated accurately. If an additional do1Iar is expected to yield at least $2.50 in present value of milk at the requested PD fat %, the dairyman is advised beneath the milk change line to increase his price constraint. If the present value of milk is less than the average price of semen, he is advised that production alone is not expected to return his investment in 10 yr at 10% interest. These formulae are adapted from (3). Finally, if no solution is obtained, the most binding constraint is listed. Usually the operator adjusts immediately the edits or goals to obtain a solution without conferring again with the dairyman. It is critical that the operator understand the function of the program relative to dairymen's desires. Journal of Dairy Science Vol. 66, No. 3, 1983
65 2
McGILLIARD AND CLAY RESULTS
Goals from dairymen have by design come only through educational meetings and residual requests. Several unsolicited requests have been received from across the country. The vast majority of MAXBULL programs have been for Holstein dairymen, although several Jersey and a few Brown Swiss, Ayrshire, and Guernsey breeders have obtained programs. Table 4 lists average goals and solutions for 196 Holstein dairymen whose requests were processed in February, 1982. These 196 include only dairymen and not individuals representing industry and education who also have tried the program. These averages relative to individual sires in the MAXBULL file are equivalent to the 90th percentile for PD milk, 60th for PD fat %, 90th for PD dollars, and 65th for PD type. Milk and fat % are negatively correlated among breeding programs, and udder support is generally the most restrictive descriptive trait although not least available among the sires. A more complete analysis of these data has been made by McGilliard and Clay (6). PROBLEMS
Maintenance of sire information is tedious, because prices must be obtained from studs, and type information from breed organizations. The Holstein Association does not distribute type information on tape to extension dairymen or dairy records processing centers.
Excessively restrictive editing and impractical goals from dairymen cause delays in returning usable solutions. On occasion solutions will contain too many bulls, several of which may represent only single units. This occurs most frequently with requests for bulls from only one stud. The problem is alleviated by modifying the request for fewer bulls or m i n i m u m units per bull. Repeatabilities are not high and, except for elimination of Low-R bulls, can only be controlled through edits. No dairyman has objected; some individuals consider this an asset relative to genetic trend. By request, a solution can be obtained for minimum groups of units, such as 4 or 10. This, however, sacrifices the exactness of the solution. REFINEMENTS
An interactive version of MAXBULL has been developed. It queries the user for information, produces breeding programs in less than 1 min, and allows immediate editing and storage of goals. For dairymen owning semen, an inventory program will adjust their goals such that MAXBULL will suggest a set of bulls to complement their inventory. Minimum average repeatability is a possible constraint which has been suppressed in favor of the Low-R concept. Unless repeatability is tied to another trait in the matrix, bulls inferior in all traits will be chosen only to increase repeatability. A common request is to maximize
TABLE 4. Averages of MAXBULL recommendations for 196 Holstein dairymen in February 1981. Goal Trait Predicted difference (PD) milk (kg) PD Fat (%) PD Fat (kg) Repeatability (%) PD Dollars PD Type (pts) Udder support (%) Rear udder (%) Fore udder (%) Teat size, placement (%) Hind legs (%) Feet (%) Price (S/unit)
X
Solution SD
--.05
.05
.58 23 20 21 14 22 22 19
Journal of Dairy Science Vol. 66, No. 3, 1983
.56 17 16 17 14 17 16 13
X
SD
789 --.05 25 67 215 .85 29 35 36 27 33 46 18
90 .04 3 6 22 .41 19 29 23 22 21 24 12
OUR INDUSTRY TODAY PD dollar value. In MAXBULL, PD dollar value is maximized at the PD fat % specified by the dairyman. Several additional features may be plausible for future MAXBULL versions. Possibilities are calving difficulty information, linear individual type including additional traits such as stature, a maximum average genetic relationship among bulls in a solution, and percentage limits on units from various studs. An option could allow maximization of variables other than PD milk, including an individualized selection index or, to minimize risk, the lower limit of a 60% confidence interval on PD milk. CONCLUSIONS
Using the MAXBULL procedure has produced objective breeding programs tailored to the goals of individual dairymen. The solutions obtain maximum PD milk superior to normal programs while maintaining balance among the other traits. Storing requests from dairymen stabilizes their goals over time and yields automatic improvement in PD xlollar averages as the quality of the bull population progresses.
653 ACKNOWLEDGMENTS
B. R. Bell, J. E. Honnette, and N. T. Jones contributed to the initial concept of selecting sires by linear programming, and J. C. Kelley developed the interactive components. REFERENCES
1 Helgren, R. E., G. E. Shook, and R. A. Schoney. 1980. A portfolio model for dairy sire selection. J. Dairy Sci. 63(Suppl. 1): 109. 2 Holstein Association. 1982. Sire summaries 1982. Volume 1. Holstein-Friesian Assoc. Am., Brattleboro, VT. 3 McGilliard, M. L. 1978. Guidelines for purchasing semen. J. Dairy Sci. 61:1680. 4 McGilliard, M. L. 1979. Relationships among genetic goals and herd breeding expenses. J. Dairy Sci. 62:85. 5 McGilliard, M. L. 1980. Informal opinions of dairymen in Virginia. 6 McGilliard, M. L., and J. S. Clay. 1983. Breeding programs of dairymen selecting Holstein sires by computer. J. Dairy Sci. 66:654. 7 Rutter, C. R., and R. E. Pearson. 1981. Breeding structure in artificial insemination Holsteins. J. Dairy Sci. 64(Suppl. 1):74. 8 Shanks, R. D., and A. E. Freeman. 1979. Choosing progeny-tested Holstein sires that meet genetic goals at minimum cost. J. Dairy Sci. 62:1429.
Journal of Dairy Science Vol. 66, No. 3, 1983