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Journal of Dairy Science Vol. 85 No. 12 3518-3525
© 2002 by American Dairy Science Association ®
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Producer Breeding Objectives and Optimal Sire Selection

P. R. Tozer* and J. R. Stokes{dagger}

* Department of Dairy and Animal Science and
{dagger} Department of Agricultural Economics, The Pennsylvania State University, University Park 16802

Corresponding author: P. Tozer; e-mail: ptozer{at}psu.edu.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
Information from an online survey of dairy producers was used to determine how important producers perceived three different objectives in the breeding problem. The objectives were: maximizing expected net merit of the progeny, minimizing the expected progeny inbreeding coefficient, and minimizing semen expenditure. Producers were asked to rank the three objectives and then to weight the importance of each objective relative to the others. This information was then used to determine weights to be used in a multiple-objective integer program designed to select individual mates for a herd of 76 Jersey cows with known genetic background and cow net merit. The results of the multiple-objective models show that rank and relative importance of producer objectives can affect the portfolio of sires selected. Producers whose primary objective was to maximize expected net merit had a range of average expected progeny net merit of $306 to $310, but the level of expected progeny inbreeding was from 6.99 to 10.45%, with a semen cost per conception of $35 to $41. For producers who selected minimizing progeny inbreeding as the primary goal in their breeding programs, the range of inbreeding was from 6.11 to 6.60%, with lower net merit range of $274 to $301 and semen expenditure of $30 to $37 per conception. One producer selected minimizing semen cost as the primary objective. For that producer’s portfolio, the semen cost was $27 per conception and net merit was $288, with a progeny inbreeding coefficient of 10.68%. The results of this research suggest that producer information and goals have a substantial impact on the portfolio of sires selected by that producer to attain these goals.

Key Words: Individual mate selection • multiple-objective integer programming • producer objectives

Abbreviation key: AHP = Analytic hierarchy process, EPIC = expected progeny inbreeding coefficient, EPNM = expected progeny net merit, MOP = multiple-objective programming, NM = net merit


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
Typically, it is assumed that dairy producers select sires that will enhance the profitability of their dairy enterprise, (Schneeberger et al., 1982; Allaire and Thraen, 1985; Rogers, 1990; Tozer and Stokes, 2001). Although this may be true, some producers may have goals within their breeding program that are not measurable directly through an economic measure, such as profit or income. However, this is not to say that productive traits do not contribute to profitability of a dairy business. Rogers (1990) partially addressed this problem through the construction of a utility index that considered productive as well as nonproductive traits (e.g., teat placement and udder depth) that do not directly contribute to the economic well-being of the dairy producer. Rogers (1990) also incorporated risk to provide some measure of the utility derived from selecting sires with relatively lower risk measured through high repeatability sires. Shanks and Freeman (1979) examined the impact that various genetic rather than economic goals had on the sires selected when trying to minimize semen cost. Although this research included the cost of semen as an economic component, the principal objective was to measure the impact noneconomic goals had on the portfolio of sires selected.

Inbreeding in dairy cattle has direct and indirect economic consequences for a dairy producer, such as reduced milk and milk component yields, lower reproduction rates, and reduced survival of offspring, (Wiggans et al., 1995; Thompson et al., 2000). However, the perception of the impacts of inbreeding could vary across breeders depending on factors within the breeder’s herd, such as the level of inbreeding and the goals and values of the dairy producer.

It may also be reasonably assumed that individual producers may have different goals in their breeding programs when compared to their peers. This may be due to many factors including psychological and sociological factors that are not easily measured. Rogers (1990) and Nash and Rogers (1996) took these factors into account when selecting a range of risk aversion factors to include in the portfolio analysis research these researchers undertook. Stokes and Tozer (2002) surveyed a group of academic professionals and breeding company representatives to determine the importance of three breeding goals: maximization of net merit (NM), minimization of the inbreeding coefficients of the progeny from a mating, and minimization of semen expenditure. The results of this survey demonstrated that perceptions of the importance of different goals vary across individuals as well as groups.

Stokes and Tozer (2002) and Tozer and Stokes (2002) utilized multiple-objective programming (MOP) to select a portfolio of sires for a hypothetical dairy producer. The weights used by Tozer and Stokes (2001) were arbitrarily selected by the researchers and were not based on any producer-provided information. Stokes and Tozer (2002) applied weights supplied by academics or breeding company representatives in the same hypothetical situation. In this research, Stokes and Tozer (2002) grouped the results and used aggregated data to estimate weights and did not consider the impact that individual weights would have on the sires selected.

Multiple-objective programming allows a decision maker to examine the tradeoffs between competing objectives (Ballestero and Romero, 1998). Also, MOP does not require explicit imposition of constraints that may limit the set of bulls from which a producer can choose (Rogers 1990). MOP implicitly constrains the feasible set through the weights assigned to each objective. Hence, MOP is less restrictive in the set of potential candidates for inclusion in the portfolio of sires that can achieve the decision-maker’s goals.

Much previous research on selection of mates has also been a form of random mating (i.e., where the selection of particular mates is not important so long as all potential females are bred and the objectives of the breeding program are achieved) (Shanks and Freeman 1979; McGilliard and Clay 1983a, 1983b; Tozer and Stokes 2001). The problem with individual mate selection is the complexity of the problem and the degree of computing power required in allocating individual sires to females and the criteria used to select mating pairs.

The objectives of the research reported here are: 1) to determine a set of absolute and relative rankings from individual producers with regard to maximizing NM, minimizing the inbreeding of the offspring from a mating, and minimizing the total expenditure on semen; 2) to estimate the consistency of the weights provided; and 3) to examine the impact that individual producer-assigned weights has on the portfolio of sires selected for breeding to each female in the herd. In the context of this research, inbreeding is defined as the expected level of inbreeding of the progeny from mating a particular sire to a particular cow.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
Analytic Hierarchy Process
The weights used in the multiple-objective models were derived from an online survey of dairy producers and the application of the Analytic Hierarchy Process (AHP) proposed by Saaty (1980). The surveyed producers responded to a questionnaire that asked them to rank the three objectives: maximizing NM, minimizing the expected inbreeding of the progeny, and minimizing semen expenditure, and then ranking the importance of each objective relative to the other two. The scale of importance is shown in Table 1Go. Demographic information was not collected from producers, as the aim was to keep the survey as simple as possible. However, demographic information may have an affect on the weights assigned.


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  		Table 1. The interpretation of the weights assigned to the relative importance of objectives in the Analytic Hierarchy Process.
 
The AHP is a two-step process; after the responses to the survey were collected, they were checked for consistency. Consistency of scores indicates whether the respondent is consistent in the ranking of objectives and the weights attached to the pairwise rankings (Saaty, 1980). Responses were deemed consistent when the consistency index was < 0.10, which is the appropriate value for a decision set of three objectives (Saaty, 1980). After the raw weights were checked for consistency, the derivation of the adjusted weights for use in the MOP models was undertaken using the PROC EIGVEC procedure of SAS (SAS, 2001). The adjusted weights used in the MOP models are the first vector of eigenvalues from the matrix of raw weights for each objective from the producer responses. The first vector of eigenvalues was then normalized such that the sum of the weights = 1 (Saaty, 1980).

Multiple Objective Breeding Model
Tozer and Stokes (2001) and Stokes and Tozer (2002) provide a comprehensive discussion of the formulation of the integer multiple-objective breeding problem. Therefore, only a brief summary of the formulation will be presented here. Various formulations of MOP depend on the degree of linearity or nonlinearity the user specifies. Stokes and Tozer (2002) show that in the sire selection problem there is very little difference between MOP formulations; hence, the following discussion will focus on the MINIMAX model. In the case of the MINIMAX model, the objective is to minimize the maximum distance, {lambda}, from the ideal values generated from the single-objective models normalized by the difference between the ideal and anti-ideal values (Ignizio, 1982).

The MINIMAX formulation is as follows:


Formula 1[1]

Subject to:


Formula 2[2]


Formula 3[3]


Formula 4[4]

x {varepsilon} F(x)

Equations 2, 3, and 4GoGoGo are distance equations that constrain the distance from an ideal solution to that found in the multiple-objective solution weighted by producer-assigned weights, wj, where j = N, I, C, (N = net merit, I = inbreeding, C = semen cost). The ideal, j* = N, I, C, and anti-ideal values, j– = N, I, C, are determined by the solution of single-objective models that optimize each of the individual objectives. The final constraint ensures that the vector of optimal values, x, needs to be within the feasible set F(x). The anti-ideal values are the maximal (minimal) value of the objective for a minimization (maximization) problem generated by the solutions for other objectives. These values measure the distance from the single-objective solution to the solution generated in the multiple-objective model and provide upper and lower bounds for the search algorithm.

Given the above, farm-level information is required. The dairy farmer has n = 76 cows, all of which are to be bred, and it is assumed that it requires, on average, two inseminations per conception. The number of bulls available is 49; these are the top bulls ranked in the Production Type Index from the American Jersey Cattle Association for which semen price, expected progeny NM (EPNM), and expected progeny inbreeding coefficients (EPIC) were available. The following three equations are accounting equations that are used to calculate the average EPNM, average EPIC, and total semen cost that are used in equations 2, 3, and 4GoGoGo.


Formula 5[5]


Formula 6[6]


Formula 7[7]

Where x(i, j) is a binary variable that represents the breeding of cow i to bull j. Bull(j) is a general integer variable representing the number of units of semen to use from bull j, and P(j) is the cost per unit of semen from bull j. The EPNM(i, j) is calculated as the average of both potential parents’ reported NM. The expected progeny inbreeding coefficient, EPIC(i, j), is calculated as the level of inbreeding of the progeny from each potential mating based on the relationships of 16 previous generations from both the sire’s and dam’s sides. The American Jersey Cattle Association provided the EPNM(i, j) and EPIC(i, j) based on the August 2001 Jersey sire summary.

One constraint imposed on the breeding formulation is a limit on the usage of any one bull to increase the potential genetic diversity within the herd.1 This limit is set at a maximum of 20%, meaning that of all cows within the herd, only a maximum of 20% can be mated to any one bull as specified in equation 8Go.


Formula 8[8]

One final constraint, necessary for completeness of the model is that cow (i) can only be bred once, so summing over all bulls gives:


Formula 9[9]

The ideal and anti-ideal values for each objective were estimated from single-objective integer programming models constrained by equations 8 and 9GoGo. Equations 5 to 7GoGoGo were utilized in the same manner as in the multiple-objective models, as accounting equations. The three single-objective models for maximizing average EPNM; minimizing inbreeding or minimizing semen cost are:


Formula 9

or


Formula 9

or


Formula 9

Subject to:


Formula 10[10]


Formula 11[11]
.

All models were estimated as integer programming models using LINGO version 7 (Schrage 2001) on a desktop personal computer. LINGO can be used in this instance as it has linear and integer programming capabilities.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 
Producer Survey Responses and Weight Calculations from the AHP
Twenty-two producers responded to the online survey, the results of which are presented in Table 2Go. The producer responses were sorted based on the rankings of each objective. The majority of producers ranked maximizing NM and minimizing inbreeding either as first or second objective. Only seven of the respondents ranked minimizing semen cost as a first or second objective. The consistency index for each producer is also presented in Table 2Go; the range of this index is from zero (0) to one (1). Typically, most producers were reasonably consistent with index scores between 0 and 0.20. However, as noted in the previous section, the critical value was established at 0.10; therefore, only 10 producers were consistent within the acceptable range.


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  		Table 2. The raw information from the producer survey, consistency index for all producers, and the adjusted weights from the AHP for producers with a consistency index ≤ 0.10.
 
Using only the information from those producers whose consistency index value was within the acceptable range, the adjusted weights were calculated for each producer and their individual preferences. The range of weights for the first objective was from 0.4667 to 0.9949. This range indicates that the first preference for some producers was not strongly preferred over the second; and for others, the first preference was extremely important. These types of results were expected because, as noted above, different producers have different sets of goals with respect to their breeding program.

Table 2Go also shows the effect of slight differences in rankings on the weights assigned to each objective. Looking at Group 1 and producers 7, 8, and 22, the raw weights given to the objectives differ across these three producers, as they perceive the importance of each objective slightly differently. Even a slight difference in raw weights can alter the adjusted weights. Producers 7 and 8 differ only on the weight assigned to the second objective; yet the adjusted weights show that for producer 8 this objective is weighted twice as high as for producer 7. Further comparisons of the impact of weights will be provided later when examination of the results of the multiple-objective models are presented.

The results of the AHP show that dairy producers do rank objectives differently and that the ordering and raw weights assigned to these rankings do have a significant influence on the final adjusted weights. Another factor that arises from the AHP is that a slight change in raw weights can affect the adjusted weights when comparing across producers.

Single-Objective Integer Models
The results of the single-objective integer models are presented in Table 3Go. The results presented in this table summarize the individual mates selected for each cow in the herd, due to the space required to present the individual data. Beginning with the model that maximizes EPNM for the individual matings, as expected, the model selected the top six bulls ranked on the basis of NM. The average EPNM of the offspring is $313, average EPIC is 10.57%, and semen cost is $41 per conception. These results also show that there is always the potential for multiple optima. For example, in this case a multiple optima occurs because the fifth and sixth ranked sires have the same EPNM value. Therefore, it is possible that any combination of these two bulls can occur in the optimal solution, as the model does not have a secondary criterion to rank the bulls. Next, considering the model that minimizes the expected inbreeding coefficients of the progeny from each breeding, the EPNM of the offspring from this model is $266, EPIC is 5.77%, and semen cost is $34. Finally, the model that minimizes semen expenditure yields an EPNM of $245, EPIC of 9.84%, and average semen cost of $20 per conception.


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  		Table 3. Semen cost (SC), net merit values (NM), net merit percentiles (NM%), average EPNM, average EPIC, and average semen costs per conception, and units of semen purchased for the single and individual producer multiple-objective models. The numbers in bold are the ideal values, and those in italics are the anti-ideal values used in the individual producer multiple-objective models.
 
The single-objective models provide the ideal and anti-ideal values for use in the multiple-objective model. The ideal values are shown in bold in Table 3Go and correspond with each individual objective. The antiideal values (in italics in Table 3Go) show the conflicting nature of the three objectives, as the model that maximizes EPNM also has the anti-ideal values or highest values of EPIC and semen cost, and the model that minimizes semen expenditure also has the lowest EPNM of any of the three single-objective models.

Multiple-Objective Integer Models
Utilizing the weights generated from the AHP and the solutions from the single objective models, 10 multiple-objective integer programming models were constructed to select the optimal sire portfolio for each producer. The results from these models are also shown in Table 3Go.

The first group of producers, those that selected maximizing NM as the primary objective and minimizing inbreeding as the second objective, are producers 4, 7, 8, and 22 in Table 3Go. The portfolio selected for producer 4 is nearly identical to that of the single-objective model for EPNM. This producer ranks maximizing EPNM highly and the other two objectives lower. The sires selected here again demonstrate the multiple optima discussed earlier. The portfolio here is slightly different to the single-objective results due to the algorithm searching in a slightly different route to achieve the optima.

The groups of sires selected for the remaining three producers in this cluster yield the same EPNM, yet the average EPIC and semen cost per conception are slightly different from each other. These differences are due to the weights placed on each objective by the individual producers. The results also show that for a small reduction in EPNM ($7), the average EPIC is reduced by 1.85 to 3.58%, and semen costs are reduced by $3 to $6, when compared to the EPNM maximizing single-objective model.

The set of sires selected for this group of producers reflects the weights assigned to the objectives. For example, sire 2, the highest cost sire, is used very minimally, and other sires, such as sire 8, with lower NM and semen costs are included. This would imply that sire 8, as well as having a lower semen cost, has a lower average EPIC compared to sire 2. This implication is supported by the results that the average EPIC for sire 2 was 10.2%, while the average EPIC for sire 8 was 5.7%. Also, for producers 8 and 22, where semen cost is not weighted as highly compared to producer 7, sire 5 is used to the limit allowed or close to the limit. Producer 8 includes 11 units of sire 15 in the portfolio, implying that this sire has a lower average EPIC than other sires not included, as this producer weights minimizing inbreeding higher than the other producers in this group. Again, this implication is supported by the results. The EPIC for sire 15, for this producer’s weights, was 5.4%.

Producers from group 3, those that ranked minimizing inbreeding first and maximizing NM second, had very diverse optimal portfolios with as many as 18 sires included in some portfolios. Beginning with producer 2, it is possible to see that the group of sires selected for this producer closely resembles that of producer 22 from group 1. This occurs because of the equivalent weights placed on the first two objectives by this producer. The sires selected for producers 13 and 14 reflect the diverse genetic makeup of the herd used in this case study and the requirements that the sires selected to breed to particular cows reduce the average EPIC as opposed to increasing the EPNM. These results again demonstrate the magnitude of the tradeoffs between objectives depends on the weights placed on the individual objectives. For example, producer 13 weighted minimizing inbreeding the highest of the four producers in this group, hence the average EPIC for this producer is lowest of the four, but then the EPNM is also the lowest. Whereas, producers 2 and 20 (who have similar weights) have higher EPNM and higher average EPIC compared to producer 13.

The final two producers, 16 and 19, were in the minority that ranked semen expenditure first or second and were consistent in their rankings. Producer 16 ranked minimizing inbreeding first and minimizing semen cost second and this ranking is reflected in the resultant portfolio. The average EPIC and EPNM for this producer is relatively low, but comparable with some producers from group three. However, semen cost is the second lowest of all multiple objective models. The sires selected in this portfolio are similar to those of producer 14, but the numbers of matings for each sire is different reflecting the reversal of the rankings of EPNM and semen cost, hence this producer prefers sires with lower semen cost rather than high EPNM.

Producer 19 was the only one to rank semen cost as the primary objective. Because of the rankings and weights applied to these rankings the portfolio selected for this producer is very different compared with most of the multiple-objective sire packages. Typically, mid-range NM sires with relatively lower semen prices, compared to the EPNM single objective models, were selected, again demonstrating the tradeoffs between objectives when multiple objectives are included in the decision model. The sire with the lowest semen price was included as expected, however, most other sires with low semen cost, i.e., $10 to $15 per unit, were not included indicating that the tradeoff between the EPNM and semen cost objectives were too high to include these sires in the optimal portfolio and, therefore, sires with higher semen costs and higher EPNM were chosen.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
ACKNOWLEDGEMENTS
 REFERENCES
 
The objectives of any breeding program are to achieve the goals of the breeder, and some of the goals of the breeder may not be economic or may include goals that are not directly measurable using some economic gauge. However, it is possible to incorporate noneconomic as well as economic objectives of dairy producer into a multiple-objective breeding model. The principal key to the success of these types of models is the set of weights applied to measure the importance of each objective to the breeder. The research reported here shows that it is possible to use objective and weight information elicited from producers to determine an optimal portfolio of sires for individual cow matings. The results also show that individual sire selection, to achieve the goals of the breeding program, is also sensitive to the weights placed on each objective by the breeder. Another result is that producers generally hold NM maximization as the primary goal in their breeding program, but other goals, such as minimizing inbreeding or reducing semen cost, are also in the producer’s mind when making breeding decisions.

In designing the survey the only objectives were to gain information regarding the ranking of the three objectives of interest; net merit, inbreeding, and semen cost, and the relative importance of these objectives. Other potential objectives in the breeding program, such as selection for nonproductive traits, were not addressed, and information was not solicited from producers regarding these types of objectives. The survey could have been designed to address this problem; however, a limitation would still be required on the number of potential objectives to allow for useful information to be collected as some of these objectives, particularly nonproductive traits, are not easily quantifiable. The AHP and multiple-objective programming could also be used to analyze risk preferences within the sire selection problem as has been suggested in other decision problems by Ballestero (2001).

One final point to note is that the results of the research are contingent on the set of available sires. In this research, only the top 50 sires ranked by PTI were utilized, and other sires with lower inbreeding values were not included in the set of potential sires (American Jersey Cattle Association, 2002). While this may not affect every portfolio selected, there may be some, particularly those that weighted inbreeding more heavily, that could change if other sires with lower inbreeding values were in the set of potential mates.


   ACKNOWLEDGEMENTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 ACKNOWLEDGEMENTS
REFERENCES
 
The authors are grateful for the financial support and for the net merit and inbreeding data from the American Jersey Cattle Association.


   FOOTNOTES
 
1 The JerseyMate program of the American Jersey Cattle Association limits the maximum number of matings to a particular sire to 15% of total matings to increase the genetic diversity within the breed (American Jersey Cattle Association, 2002). Back

Received for publication January 18, 2002. Accepted for publication March 28, 2002.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 


Allaire, F. R., and C. S. Thraen. 1985. Prospectives for genetic improvement in the economic efficiency of dairy cattle. J. Dairy Sci. 68:3110–3123.[Abstract/Free Full Text]

American Jersey Cattle Association. 2002. http://www.usjersey.com/Bulls/siremgs.htm. Accessed Feb. 27, 2002.

American Jersey Cattle Association. 2002. http://www.usjersey.com/Programs/JerseyMate.pdf. Accessed Feb. 27, 2002.

Ballestero, E. C., and C. Romero. 1998. Multiple Criteria Decision Making and its Applications to Economic Problems. Kluwer Academic Publishers, Boston, MA.

Ballestero, E. C. 2001. Stochastic goal programming: A mean-variance approach. Eur. J. Operational Res. 131:476–481.

Ignizio, J. P. Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ.

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.

Nash, D. L., and G. W. Rogers. 1996. Risk management in herd sire portfolio selection: A comparison of rounded quadratic and separable convex programming. J. Dairy Sci. 79:301–309.[Abstract]

Rogers, G. W. 1990. A utility function for ranking sires that considers production, linear type traits, semen cost, and risk. J. Dairy Sci. 73:532–538.[Abstract]

Saaty, T. L., 1980. The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill, New York, NY.

SAS Institute, 2001. SAS Users Guide, Version 8.2. SAS Institute, Cary, NC.

Schneeberger, M., A. E. Freeman, and M. D. Boehlje. 1982. Application of portfolio theory to dairy sire selection. J. Dairy Sci. 65:404–409.[Abstract/Free Full Text]

Schrage, L. 2001. LINGO: The Modeling Language and Optimizer. 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.[Abstract/Free Full Text]

Stokes, J. R., and P. R. Tozer. 2002. Sire Selection with Multiple Objectives. Agric. Sys. 73:147–164.

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.[Abstract]

Tozer, P. R., and J. R. Stokes. 2001. Using multiple-objective programming in a dairy cow breeding program. J. Dairy Sci. 84:2782–2788.[Abstract]

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.[Abstract]

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.[Abstract]


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