J. Dairy Sci. 84:2782–2788 American Dairy Science Association, 2001.
Using Multiple Objective Programming in a Dairy Cow Breeding Program1 P. R. Tozer*,2 and J. R. Stokes† *Department of Dairy and Animal Science and †Department of Agricultural Economics and Rural Sociology, The Pennsylvania State University, University Park, PA 16802
ABSTRACT
INTRODUCTION
Multiple-objective programming was used to examine the effects various objectives had on the optimal portfolio of sires chosen for a given breeding problem in a Jersey cow dairy herd. It was assumed that the dairy producer had the following three objectives in the breeding decision: to maximize Net Merit, to minimize inbreeding, and to minimize total expenditure on semen. Integer programming models of these three single objectives were estimated to provide the ideal and antiideal values for use in several multiple-objective programming models. The integer multiple-objective models examined the interactions and costs of tradeoffs between the three single objectives in a model framework designed to minimize the maximum deviations from the single-objective optima. A model with equal weights on each objective resulted in a decrease of 3% in average inbreeding but also reduced average Net Merit by $170 from the single-objective optima. A second model, where the weight on Net Merit was twice that of inbreeding and semen cost, decreased Net Merit by $100 and reduced inbreeding by 2% from the single objective optima. The results of the multiple-objective programming models show that reducing the inbreeding coefficient for a group of sires purchased will decrease the Net Merit. However, the results generated also demonstrate that the weights placed on each objective by the dairy producer substantially affect the optimal levels of each objective within the multiple-objective model. (Key words: breeding, mathematical programming, Jersey, multiple objective)
The goal of any dairy breeding program is to increase the profitability of the enterprise. To achieve this goal, producers need to select animals that have the genetic potential to increase the yield of the most economically important product (milk or milk fat or milk protein) or reduce the costs caused by trait faults, such as poor conformation. One major problem with selecting sires with high genetic potential is that the sires used are likely closely related to the female(s) to be bred, which could lead to inbreeding problems in the longer term. Inbreeding is the mating of related animals, and one measure of inbreeding is the inbreeding coefficient, which measures the degree of relationship between parents; in the context of most studies discussed below and the research reported, the inbreeding coefficients are for the prospective offspring of a mating. Inbreeding in dairy cows leads to reductions in lactation performance (both in total production and in component yields), reduced survival rates for highly inbred offspring, and increased reproduction problems, which reduces the longevity of the offspring in the herd (Thompson et al., 2000). Wiggans et al. (1995) estimated that inbreeding in the U.S. Jersey breed reduced milk production by 21.3 kg, fat yield by 1.03 kg, and protein by 0.88 kg, each lactation for each 1% of inbreeding. Miglior et al. (1992), by using Canadian Jersey data, predicted a reduction of 9.84 kg of milk, and 0.5 kg of fat per lactation per 1% of inbreeding. Wilk and McDaniel (1995, 1996) examined the survival of the offspring of inbred and noninbred dams and determined that inbreeding levels greater than 1% reduced the probability of survival to first calving. Hence, inbreeding does have immediate and long-term consequences on the profitability of the dairy breeding program and the dairy business. Mathematical programming has been used in previous research to determine the optimal breeding strategy for dairy producers. In the mathematical programming formulation, the objective is the function to be maximized or minimized, and a constraint restricts the set of possible results due to various factors, such as
Abbreviation key: MOP = multiple objective programming, NM = Net Merit.
Received December 26, 2000. Accepted July 18, 2001. Corresponding author: P. R. Tozer; e-mail:
[email protected]. 1 This research was a component of NC-119, Dairy Herd Management Strategies for Improved Decision Making and Profitability.
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resource limits or minimum input requirements. Shanks and Freeman (1979) applied linear programming in the selection of sires to meet various genetic goals of a dairy breeder at minimum semen cost to the breeder. McGilliard and Clay (1983a,b) used linear programming in the MAXBULL program to maximize the predicted difference in milk while constraining variables such as minimum udder conformation traits, maximum semen price, maximum units per sire, and maximum number of sons per sire. This last variable could be considered a proxy for reducing potential inbreeding caused by having too many sires from the same breeding line. A similar approach was taken by Jansen and Wilton (1984, 1985), but the objective in that research was to maximize a profit function based on a set of estimated breeding values for milk and fat, and the costs and returns for feed, labor, milk, and fat. These researchers recognized the long-term consequences of sire selection on both the economic and genetic structure of the dairy herd but did not explicitly incorporate inbreeding or some measure of inbreeding into their decision model. The use of inbred sires by a dairy producer is not a problem in the breeding decision if the sires are to be mated to unrelated potential mates. Nash and Rogers (1995) used portfolio theory to examine the sire selection problem. The objective was to maximize the expected net revenue of the sire portfolio, subject to constraints on maximum average price for semen, a minimum number of total units of semen, a minimum number of sires that transmit calving ease for breeding heifers, and a minimum on the reliability for the evaluations of milk. One of the first breeding optimization models to explicitly consider inbreeding into the decision process was developed by Weigel and Lin (2000). These researchers maximized Net Merit (NM), subject to a given level of progeny inbreeding (i.e., 8, 9, or 10% in Jerseys). Another constraint in the model to help control the undesirable effects of inbreeding was a maximum number of matings to any one sire. In all the studies discussed above there was only one objective, to maximize some measure of production or economic return, i.e., milk production, profit, or NM. However, it would be reasonable to assume that dairy producers have more than one objective in mind with respect to their breeding programs. Although some measure of economic returns may be a primary goal of the program, reducing the inbreeding coefficients of the progeny and minimizing or controlling expenditure on semen are also likely objectives. Multiple-objective programming (MOP) has been used infrequently in animal agriculture. Many of the applications of MOP have been in ration formulation problems (see, for example, Lara, 1993; Rehman and Romero, 1984, 1987; Tozer and Stokes, 2001). This technique has a similar structure
to the single-objective models, such as those discussed in the previous paragraph; however, the objective function is flexible and allows the incorporation of more than one objective. Galligan et al. (1988) used multiobjective linear programming in sire selection and concluded that MOP can be used to achieve a set of balanced breeding objectives faster than can be reached by using single-objective techniques. The objective in this study was to demonstrate an application of integer multiple-objective programming to the general breeding problem for a dairy producer with a herd of Jersey cows and to explicitly consider the effect inbreeding has on the decision process. In the model we are assuming the dairy producer has the following three competing objectives: to maximize NM, to minimize the inbreeding of the progeny, and to minimize the total expenditure on semen. Other objectives could be added to the problem, such as selection for particular traits: however, these types of objectives are more farm and decision-maker specific and would remove us from the generality of the process without adding substantially to the content of the research. However, this is not to say that these objectives are unimportant to individual decision makers. MATERIALS AND METHODS Multiple-objective programming seeks solutions that are consistent with multiple-competing objectives. Each objective has a weight attached to it and the weight measures the importance to the decision maker of the objective relative to the other objectives. One method of solving multiple-objective programming models is to use a MINIMAX approach. The intent of the MINIMAX method is to minimize the maximum deviation from the ideal or target value identified for each objective. In MINIMAX form the basic MOP model can be written as: minimize λ
[1]
Subject to: fi*
wi
− fi(x) ≤ λ, gi
∀i = 1, . . ., N
x⑀F
[2] [3]
Here, λ measures the maximum distance the solution is from the ith objective’s ideal value where there are N objectives. The weight wi attached to each objective is determined by the decision maker, f*i is the best value for each objective, fi(x) is the value of the objective determined within the MOP model. The gi are the normalizJournal of Dairy Science Vol. 84, No. 12, 2001
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ing constants or degradation allowances. These normalizing constants can take many forms, either an algebraic function or a simple constant depending on the information available. Equations of the form of equation 2 must be normalized or scaled to ensure that all objectives or values are dimensionless. If normalizing were not done, objectives with relatively large values will usually dominate those with smaller values, even if the objectives with smaller values were weighted higher. Equation 3 simply states that the vector of optimal values (x) must be within the feasible set F. The normalizing constant can be specified in many ways; one such way is gi = f*i − f∼i where f∼i is the worst value for the ith objective. In the multiple-objective breeding problem, the best values are likely unobtainable and are defined as maximal Net Merit (N), minimal semen cost (C), and minimal inbreeding percentage (I), where inbreeding is defined as the expected inbreeding coefficient of the progeny. That is, maximal NM likely does not induce minimal semen cost or inbreeding percentage and the converse holds for the other variables. By use of the above, the MOP MINIMAX specification for the breeding problem can be written as follows: minimize λ
[4]
Subject to: − N(x) ≤ λ N* − N ∼
N*
wN
I* − I(x) wI I* − I∼
m
Max ≤λ
− C(x) ≤ λ C* − C∼
C*
wC
[5]
The constraints specified by equations 5, 6, and 7 measure the weighted percentage deviations from the best values for each objective. The three equation numbers 8, 9, and 10 are accounting equations to measure the NM, average inbreeding percentage, and total cost of semen. The parameters nj, bj, and cj are the NM, inbreeding coefficient of potential progeny, and cost per unit of semen, respectively, for sire j. The NM values are assumed to be independent of the effects of inbreeding (VanRaden, 2000). The parameter m is the total number of potential sires available, in this study m = 98. Equality constraint, equation 11, ensures that only the required amount of semen (R) is purchased, R is determined by the decision maker before sire selection. Equation 12 places a maximum on the number of units of semen to be used from any one sire. This maximum is determined by the value of k (0 < k < 1). The final constraint, equation 13, imposes an integer value on the number of units of semen that can be purchased, as it is not possible to buy fractions of units of semen. The best values for each objective, N*, C*, or I*, are determined by the solution of independent single-objective models for each objective. The worst values, N∼, C∼ or I∼, are also determined within the single-objective models. Hence, we will have the following three sets of values for each model: an NM total, a total cost of semen, and an average inbreeding coefficient. The three objective functions for the single-objective models can be written as follows:
∑njxj/R
∑cjxj /R
[7] Min [8]
or m
[9]
[10]
j=1
m
∑xj = R
[11]
0 ≤ xj ≤ k ∗ R
[12]
xj = integer ∀ j = 1, . . ., m
[13]
j=1
Journal of Dairy Science Vol. 84, No. 12, 2001
[14b]
or
m
C(x) =
m I = ∑bjxj/R j=1
[6]
j=1
m I(x) = ∑bjxj /R j=1
[14a]
j=1
m
N(x) =
∑njxj/R
N=
Min
C=
∑cjxj/R
[14c]
j=1
Equations 14a, 14b, and 14c are constrained by equations 11 through 13. Given the above model specification, the data requirements are relatively limited. Data are required for the NM scores, expected inbreeding coefficients of the progeny, unit semen costs, the total units of semen required, and the percentage of matings to any one sire (i.e., the value of k). NM was derived from the August 2000 Sire Summaries. The number of Jersey sires in the August sire summary was 98; of these sires, 86 had expected
MULTIPLE OBJECTIVE BREEDING PROGRAMS
inbreeding coefficients reported by the American Jersey Cattle Association (2000) because they were domestically bred sires. The remaining 12 sires were either imported or their semen was imported and the expected progeny inbreeding coefficients were available from the Animal Improvement Projects Laboratory (AIPL) (2000). The expected progeny inbreeding coefficients are based on a random mating to 600 potential mates that are assumed to be representative of the Jersey breed (American Jersey Cattle Association, 2000). The expected inbreeding coefficients for domestic and imported sires are based on the same method of estimation. Per-unit semen costs were provided through the National Association of Animal Breeders (American Jersey Cattle Association, 2000). In this study, the weights wN, wI, and wC are determined arbitrarily by the researchers and are set at one, two, or three. However, weights could be elicited from individual dairy producers based on how the producer perceives the relative importance of each objective and an application of the analytic hierarchy process so the producer does not have to directly assign weights to each objective (Saaty, 1980). For demonstration purposes, we are assuming that the dairy producer has a herd size of 60 cows to be bred every year and that the dairy producer purchases semen four times per year. Hence, of the 60 cows and heifers, 25% or 15 are bred each quarter, with an average number of inseminations per conception of 2.5 (Lancaster D.H.I.A., 2000). By using these figures, the dairy producer must purchase 38 units of semen per quarter. In this study, the value of k is set at 0.2, indicating that the maximum number of units of semen from any one sire is 0.2 × 38 = 7.6. However, because of the integer constraint imposed by equation 13, the value for the maximum number of units of semen that can be purchased is truncated to the lower integer value, in this case, seven. All models were estimated as integer programming models by use of LINGO version 6 (Schrage, 2000) on a desktop personal computer. LINGO can be used in this instance because it has linear and integer programming capabilities. RESULTS AND DISCUSSION The results of the single-objective models are presented in Table 1. From these single-objective model results, it is possible to see the competitive nature of the objectives, which is particularly apparent when comparing the NM maximization model with the inbreeding minimization model. The NM maximization model yields an average of $435 (98th percentile; the percentiles are for each trait are based on all sires in
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the data set) and an average inbreeding level of 7.41% (48th percentile), which is the highest of the three single-objective models. Conversely, the inbreeding minimization model yields an average NM of $117 (10th percentile), the lowest of the three single-objective models and an average inbreeding coefficient of 1.6% (5th percentile). The average semen cost is also highest in the NM model, again demonstrating the competitive nature of the NM maximization and cost minimization objectives. The numbers in boldface in Table 1 are the best values required for the multiple-objective model. The italicized values are the antiideal or worst values the singleobjective models can take. Table 1 also reports the semen price, NM, and expected inbreeding coefficient of progeny for the sires selected within either the singleobjective or the multiple-objective models. All 98 sires were potential candidates for inclusion in either the single- or multiple-objective models; however only sires selected within any model are included in the results. Data for sires not selected by any model are not included because of space limitations but are available from the authors. The sires selected by each single-objective model follow the expected pattern; sires with high NM are chosen in the NM maximizing model, sires with low semen cost are selected in the cost minimization model, and sires with low progeny inbreeding coefficients are picked in the inbreeding minimization model. One feature of sire selection is that each model is mutually exclusive of the others; that is, no sire appears in more than one portfolio. This set of results is based on the vectors of NM, inbreeding coefficients, and semen costs used; however, the results could change with a different set of vectors. Given the best and worst values are identified, it is possible to consider the multiple-objective function models. Table 1 also summarizes the results of several multiple-objective models. Model 1:1:1 in Table 1 is an equal-weight multiple-objective model; i.e., the decision maker believes that all three objectives are equally important. By comparing the single-objective models to this equal-weight model, it is possible to examine the tradeoffs between objectives when entered into a multiple-objective framework. The average cost of semen in the multiple-objective model is $14.18; this represents an increase of 151% from the cost minimization model, but a reduction of 35% and 10% in the cost of semen from the NM and inbreeding models, respectively. Similar tradeoffs occur when comparing the NM and average inbreeding percentages for the other two models. By comparing the NM single-objective results to the equalweight multiple-objective model, it is apparent that there is reduction in the levels of all three variables of Journal of Dairy Science Vol. 84, No. 12, 2001
Journal of Dairy Science Vol. 84, No. 12, 2001
26 20 20 18 25 20 22 18 5 20 7 10 6 15 8 16 6 5 16 16 16 16
Bull No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
481 442 432 417 416 402 381 360 344 341 279 270 231 206 193 181 171 156 155 143 65 (10)
USDA Net Merit ($) 7.0 7.0 8.2 7.3 7.3 8.0 6.4 5.4 9.1 4.7 5.2 6.4 7.3 2.2 5.2 3.8 7.3 5.4 1.3 1.2 1.2 1.0
Expected inbreeding coefficient (%)
1 7
6
7 3 7
7
7 7 7 7 7 3
Units of semen purchased
21.663 435.00 7.41
6.452 242.34 6.49 Units of semen purchased
Maximize Net Merit
Minimize cost
7 7 7 7
3
7
Units of semen purchased
15.82 117.00 1.60
Minimize inbreeding
7 2
7
7
5 4
6
Units of semen purchased
13.89 277.00 4.47
Model 1:1:11
2 5
7 5
7
5 7
Units of semen purchased
10.76 254.08 4.89
Model 2:1:1
2
1
7
7 1 3 7
3
2 7
Units of semen purchased
15.97 335.37 5.24
Model 1:2:1
Ratios indicate weights on semen cost, Net Merit, and inbreeding levels, respectively, in the MINIMAX formulation. Boldfaced values indicate ideal values for use in multiple-objective models. 3 Italicized values indicate antiideal values for use in multiple-objective models.
1
Semen price ($/unit)
Average cost per unit of semen ($) Average Net Merit ($) Average inbreeding (%)
Objective
7 7
6
4 6
7
1
Units of semen purchased
15.47 246.24 3.30
Model 1:1:2
7 7
7
7
1 1
7 1
Units of semen purchased
18.71 275.39 3.52
Model 1:2:3
Table 1. Semen prices, Net Merit values, inbreeding coefficients, results, and units of semen purchased for the single- and multiple-objective models.
7 7
7
7
1 1
7 1
Units of semen purchased
19.29 327.76 4.52
Model 1:3:2
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approximately 35 to 40%. In the sense of reducing costs and inbreeding, this result is desirable; however, these reductions come at a cost of a reduction in NM, which is less attractive to dairy producers. The largest relative change occurs when comparing the inbreeding single-objective model to the equalweight multiple-objective results. In this model, the average cost of semen falls by 12%, from $15.82 to $14.18, but NM increases by 140%; this increase in NM is offset by an increase in the inbreeding percentage of 179%. These changes must be considered in the context of the levels determined within the single-objective model. The inbreeding percentage in the single-objective model is 1.6% (5th percentile) compared with a level of 4.41% (8th percentile) in the multiple-objective model. However, NM increases from an average of $117 (10th percentile) to $281 (54th percentile). The level of inbreeding in the equal-weight multipleobjective model is less than that reported by Weigel and Lin (2000) for the unconstrained or inbreeding constrained models. It is not possible to compare the NM values generated in this research with those of Weigel and Lin (2000), because the method of calculating NM differs and also sire NMs are reestimated each time a new sire summary is released. Previously, NM was based on an annual value but is now based on expected lifetime performance (VanRaden, 2000). Changing the relative weights of each objective impacts the value of the three objectives significantly. For example, when looking at the average expenditure on semen across Table 1, the range is from $10.97 (25th percentile) to $19.29 (88th percentile). The average cost of semen is affected by the relative weight assigned to this objective; when cost is twice as important as the other two objectives, the cost is at the low end of the range. When the expenditure on semen is weighted relatively low, then the average cost of semen is relatively high. Changing the weights assigned to each objective also changes the sires that are included in the portfolio, which is as expected. For portfolios where the weight assigned to cost is relatively high, sires with low semen cost are included in the package. However, as cost is assigned a relatively lower weight, the number of these low-cost sires diminished and sires that satisfy the objectives with higher weights replace them. Some sires appear in most portfolios; this is due to the relative values of the three objectives these sires possess when compared with sires excluded. For example, sires 14 and 19 appear in all the portfolios because of the relatively low semen price per unit and low inbreeding value. Although these sires have a low NM value, the relative value provided by the other two variables out-
weighs the value that could be provided by alternative sires. When comparing the sires selected in the multipleobjective models with those in the single-objective models, it is apparent that some sires do not satisfy the various weighted multiple objectives. In the single objective models, 18 different sires were selected to satisfy one of the three objectives. However, in the multipleobjective models, only 13 different sires are selected and, of these 13 sires, four are not included in any of the single-objective portfolios. Again, the reason for this change in sire selection is the effect the competing objectives have on the relative value of each of the sire’s variables of cost, NM, and inbreeding coefficient. Multiple-objective programming provides some detail on the magnitude of the tradeoffs between objectives. These tradeoffs can be quantified in the short term, but long-term quantification is more difficult. For example, use of the reduction in milk production of 21.3 kg per percentage point of inbreeding, from Wiggans et al. (1995), and the highest and lowest inbreeding percentages from the multiple-objective models, 5.18 and 3.28%, yields a loss of milk production of approximately 42.6 kg per lactation. However, the impact of inbreeding and the loss of genetic variation through inbreeding is a longer term phenomenon; hence, the longer term effects may be much greater, depending on the objectives of the dairy producer. CONCLUSIONS The objectives of any breeding program are to increase the profitability of the enterprise of interest and, as such, dairy cattle breeding programs must consider the multiple objectives necessary to increase and sustain the profitability of the dairy business. In this research, it has been demonstrated that designing breeding programs with multiple objectives can achieve the multiple goals of the program. However, achievement of these goals comes at a cost when compared with a single objective. Multiple-objective programming was used to examine the tradeoffs between total expenditure on semen, NM, and the inbreeding coefficients of a dairy producer breeding Jersey cows. The determining factor in the relative size of the tradeoffs is the weight the dairy producer places on each objective. A traditional method of sire selection has been the use of selection indexes; multiple-objective programming can be used in conjunction with selection indexes to allow a dairy producer to assign higher weights to those traits that are more desirable for the producer and to select sires that may be more suited to the goals of the breeder. Multipleobjective programming is not a different method of sire Journal of Dairy Science Vol. 84, No. 12, 2001
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selection but a tool to aid breeders in selecting sires that may be more suited to their goals. Two final factors must be considered when researching the breeding decision in the context of the research presented here. The first aspect of the breeding decision that was not considered was the temporal dynamics of the breeding decision and how these dynamics affect farm profitability over a longer period of time. The second factor that must be acknowledged is the uncertainty or risk inherent in all breeding decisions. In this study, a measure of risk was not included in the models presented because of the nature of the models constructed and the lack of adequate data to capture the risks in the breeding decision. ACKNOWLEDGMENTS The authors would like to thank the American Jersey Cattle Association for financial support. REFERENCES American Jersey Cattle Association. 2000. http://www.usjersey.com/ Bulls/siremgs.htm. Accessed September 21, 2000. Animal Improvement Projects Laboratory (AIPL). 2000. http:// aipl.arsusda.gov/. Accessed September 21, 2000. Galligan, D. T., C. F. Ramberg, W. Chalupa, C. Johnstone, and G. Smith. 1988. An application of multi-objective linear programming for selection of sires. J. Dairy Sci. 71(Suppl. 1): 237. (Abstr.) Jansen, G. B., and J. W. Wilton. 1984. Linear programming in the selection of livestock. J. Dairy Sci. 67:897–901. Jansen, G. B., and J. W. Wilton. 1985. Selecting mating pairs with linear programming techniques. J. Dairy Sci. 68:1302–1305. Lancaster D. H. I. A. 2000. October Report. Lancaster D. H. I. A., Manheim, PA. Lara, P. 1993. Multiple objective fractional programming and livestock ration formulations: A case study for dairy cow diets in Spain. Agric. Syst. 41:321–334.
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McGilliard, M. L., and J. S. Clay. 1983a. Breeding programs of dairymen selecting Holstein sires by computer. J. Dairy Sci. 66:654– 659. McGilliard, M. L., and J. S. Clay. 1983b. Selecting groups of sires by computer to maximize herd breeding goals. J. Dairy Sci. 66:647–653. Miglior, F., B. Szkotnicki, and E. B. Burnside. 1992. Analysis of levels of inbreeding and inbreeding depression in Jersey cattle. J. Dairy Sci. 75:1112–1118. Nash, D. L., and G. W. Rogers. 1995. Herd sire portfolio selection: A comparison of rounded linear and integer programming. J. Dairy Sci. 78:2486–2495. Rehman, T., and C. Romero. 1984. Multiple-criteria decision-making techniques and their role in livestock ration formulation. Agric. Syst. 15:23–49. Rehman, T., and C. Romero. 1987. Goal programming with penalty functions and livestock ration formulation. Agric. Syst. 23:117– 132. Saaty, T. L. 1980. The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill, New York, NY. Schrage, L. 2000. Optimization Modeling with LINGO. LINDO Systems, Chicago, IL. Shanks, R. D., and A. E. Freeman. 1979. Choosing progeny-tested Holstein sires that meet genetic goals at minimum semen cost. J. Dairy Sci. 62:1429–1434. Thompson, J. R., R. W. Everett, and C. W. Wolfe. 2000. Effects of inbreeding on production and survival in Jerseys. J. Dairy Sci. 83:2131–2138. Tozer, P. R., and J. R. Stokes. 2001. A multi-objective programming approach to feed ration balancing and nutrient management. Agric. Syst. 67:201–215. VanRaden, P. 2000. Net Merit as a Measure of Lifetime Profit. http:// aipl.arsusda.gov/html/nm2000.html. Accessed September 26, 2000. Wiggans, G. R., P. M. VanRaden, and J. Zuurbier. 1995. Calculation and use of inbreeding coefficients for genetic evaluation of United States dairy cattle. J. Dairy Sci. 78:1584–1590. Wiegel, K. A., and S. W. Lin. 2000. Use of computerized mate selection programs to control inbreeding of Holstein and Jersey cattle in the next generation. J. Dairy Sci. 83:822–828. Wilk, J. C., and B. T. McDaniel. 1995. Influence of genetic diversity on viability in Randleigh Jerseys. J. Dairy Sci. 78(Suppl. 1):154. (Abstr.) Wilk, J. C., and B. T. McDaniel. 1996. Effect of inbreeding on heifer survival to first calving in Jerseys. J. Dairy Sci. 79(Suppl. 1):205. (Abstr.)
