Introduction

Genomic selection (GS) is a promising method with high potential to improve selection of complex traits like grain yield (He et al. 2016). Most research on GS has focused on prediction models and prediction accuracies in different crops (Bernardo and Yu 2007; Jannink et al. 2010; Heslot et al. 2012; Windhausen et al. 2012; Lorenz 2013; Technow et al. 2014; Lehermeier et al. 2014; Zhang et al. 2015; Zhao et al. 2015). Although different prediction models were applied, they achieved very similar prediction accuracies. In hybrid breeding, three prediction scenarios must be distinguished: prediction of hybrids for which no (T0), one (T1) or both parents (T2) have been phenotyped for their general combining ability (GCA) of grain yield (Technow et al. 2014; Zhao et al. 2015). Thereby, T0 represents the prediction of new DH lines based on calibrations developed in previous cycles of selection (Longin et al. 2015). Current studies in hybrid wheat, maize, rice and rye report average T0—prediction accuracies of GS for grain yield varying from 0.2 to 0.4 (Zhao et al. 2013; Albrecht et al. 2014; Bernal-Vasquez et al. 2014; Xu et al. 2014). For hybrid barley and triticale, we did not find any report in the literature on GS prediction accuracies.

Only one report assessing the optimum allocation of resources maximizing selection gain in multi-stage breeding strategies using GS is currently available (Longin et al. 2015). In this study, GS and classical phenotypic selection were compared based on the expected annual selection gain for hybrid wheat. Assuming a single scenario with fixed budget, variance components and costs of doubled haploid (DH) line and hybrid seed production, a breeding strategy combining one-stage GS followed by one-stage phenotypic selection was optimum across a wide range of GS prediction accuracies. However, generalization of these results is difficult, because budgets, costs and variance components may vary largely across breeding programs and especially across crops. Additionally, this previous work assumed that all produced DH lines directly enter genotyping or field testing for grain yield. In applied cereal breeding, however, new DH lines are first evaluated in small observation plots before expensive genotyping or yield phenotyping (Cooper et al. 2014; Bassi et al. 2015). Selection among candidates at this stage is based on their per se performance regarding highly heritable traits, e.g., plant height, disease resistance and heading time (cf. Longin et al. 2013). Only the best performing DH lines enter genotyping, hybrid seed production and yield trials. The implementation of nursery selection in optimization studies might largely change the efficiency of breeding strategies and their optimum allocation of resources, but it has not been investigated.

We, therefore, expanded the open source R package “selectiongain” (Mi et al. 2014) to compare hybrid breeding strategies including nursery selection, GS, and phenotyping for various scenarios regarding variance components, budgets and costs of DH line and hybrid seed production. Our objectives were to (1) determine the effect of nursery selection on the annual expected selection gain of several breeding strategies, (2) investigate the impact of budget, costs and variance components on the efficiency of several breeding strategies and their respective optimum allocation of resources, and (3) determine most efficient hybrid breeding strategies for the improvement of grain yield in wheat, maize, rye, barley, rice and triticale.

Materials and methods

Breeding strategies

For all breeding strategies, we assumed the production of N ini DH lines from the cross of numerous inbred parents. After DH line production, a number of N ini DH lines are tested in an observation nursery for their per se performance regarding highly heritable traits, e.g., disease resistance, heading time, plant height. The proportion of lines selected after nursery assessment (α) was evaluated in one to three stages of general combining ability (GCA) performance for grain yield. In all GS strategies, the first selection for grain yield was based on GS across N 1 DH lines selected in the nursery. For the breeding strategy GSonly, no further phenotypic selection was made, representing one-stage selection based on markers only. This is the fastest possible breeding strategy with GS. In contrast, for the other strategies, GS was followed by one-stage (GSrapid) or two-stage phenotypic selection (GSstandard) for grain yield with N 2 and N 3 candidates, respectively. To assess the advantage of GS over breeding strategies based only on phenotypic data, we used hybrid breeding strategies PSrapid and PSstandard, i.e., one or two-stage phenotypic selection without GS in the first stage. PSstandard represents the conventional practice in hybrid cereal breeding programs and was considered in our treatise as the standard for comparisons. To make strategies directly comparable, we assumed that a number of five lines are finally selected in each different breeding strategy. Thus, the estimation of genetic gain for all strategies is based on the same final number of five selected lines (Fig. 1). Following this selection, the best testcrosses of these lines could proceed to further field evaluation in (pre)registration trials and, additionally, the five selected lines could be evaluated for potential commercial use in new hybrid combinations. However, these further stages of hybrid testing are outside the scope of our study.

Fig. 1
figure 1

Hybrid breeding strategies with nursery selection on an initial number of \(N_{\text{ini}} = \frac{{N_{1} }}{\alpha }\) DH lines from numerous crosses, assuming that traits selected in the nursery are not correlated with the target trait grain yield. After nursery, the first stage of grain yield selection is based on GS followed by four different ways to assess grain yield performance in the field with their respective allocation of test resources. In stages 2 and 3, phenotypic selection is based on GCA and selection is performed after the respective field trials. In breeding strategies “rapid”, the second field test is omitted speeding up the breeding process. For strategy “GSonly”, only genomic selection is performed without any further field test, minimizing the breeding cycle length. In all breeding strategies, a final number of five lines was selected for evaluation of potential commercial use in new hybrid combinations. For comparison, standard hybrid breeding strategies without genomic selection were also optimized (PS standard and PS rapid ) (N j , L j , T j number of DH lines, test locations and testers in stages one, two and three, respectively)

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For simplicity, we assumed the use of a chemical hybridization agent (CHA) for hybrid seed production of wheat, rice, triticale, barley and rye. In the case of maize, we assumed pollination in isolation fields. However, for barley and rye the successful use of a CHA for hybrid seed production is yet not possible. Hybrid seed production for these crops is currently based on cytoplasmic male sterility (CMS). As intensively described at the end of the discussion, our results can still be applied for the male pools of these CMS breeding programs, i.e., the maintainer and restorer groups. However, further research is necessary to investigate best breeding schemes for the female pool of CMS breeding programs. For phenotypic selection, the testcross performance was evaluated at L j locations with T j testers in selection stage two and three (j = 2, 3). We assumed the use of inbred lines as testers. Without restrictions on L j , the selection gain (ΔG) is maximized for one replication per location (Melchinger et al. 2005). Thus, we set the number of replications to one for all calculations. An overview of the abbreviations used throughout the manuscript is given in Table 1.

Table 1 Scenarios and abbreviations used in this manuscript
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Calculation of the selection gain

Calculation of ΔG is based on the well-known formula of Cochran (1951) with multivariate normal integrals for selected fractions and heritabilities. The five investigated breeding strategies differ up to 4 years regarding their cycle length (Fig. 1). To account for this difference, we also determined the annual selection gain ΔG a, i.e., the absolute ΔG divided by the number of years required to complete the respective breeding strategy. For all our calculations, we used the open source R (R Development Core Team 2016) package “selectiongain”, which was extended for the incorporation of nursery selection in the computation of the expected selection gain and the optimum allocation of resources. For details, the reader is referred to Mi et al. (2015) and the user manual of the package version 2.0.50.1, which now includes nursery selection (http://cran.r-project.org/web/packages/selectiongain/selectiongain.pdf).

Optimum allocation of resources

After nursery evaluation of N ini DH lines, N 1 lines was selected to perform GS on grain yield. In stage two and three, N j refers to the number of DH lines tested at L j locations with T j testers. A vector (N ini, N 1, N j , L j , T j ) is denoted as optimum allocation if it maximizes the selection gain (ΔG) in the respective breeding strategy for a given scenario regarding variance components, budgets and costs. The optimum allocation was determined by a grid search across all possible allocations for the scenario under consideration (for details see Mi et al. 2015).

Quantitative genetic parameters and economic frame

Variance components were taken from a large experimental study of wheat hybrids, (Longin et al. 2013), with \({ \sigma_{{GCA}}^{2}} = 5.7\), \({\sigma_{{{{GCA }} \times L}}^{2}} = 5.19\), \({\sigma_{{SCA}}^{2}} = 1.88\), \({\sigma_{{{{SCA }} \times L}}^{2}} = 2.94\), and \({\sigma_{{error}}^{2}} = 24.37\), where \( \sigma_{{GCA}}^{2}\), \( \sigma_{{SCA}}^{2}\), refer to the variances of GCA and specific combining ability (SCA), respectively, \( \sigma_{{{{GCA}} \times L}}^{2}\) and \( \sigma_{{{{SCA}} \times L}}^{2}\) to the variances of the interaction of these combining abilities with locations, and \( \sigma_{{error}}^{2}\) to the variance of the plot error (Table 1). This ratio of variance components was used as standard and is in the following referred to as variance component scenario 1, representing wheat (DH lines) and rye (S2 lines) hybrid breeding (Longin et al. 2013; Bernal-Vazquez et al. 2015). To investigate the influence of different variance components on the optimum allocation of test resources and the ranking of different breeding strategies, we used two further variance component scenarios. In scenario 2, \( \sigma_{{SCA}}^{2}\) was doubled representing roughly variance components from maize (Longin et al. 2007). In scenario 3, \( \sigma_{{SCA}}^{2}\), \( \sigma_{{{{GCA }} \times L}}^{2}\), \( \sigma_{{{{SCA}} \times L}}^{2}\) and \( \sigma_{{error}}^{2}\) were doubled compared to scenario 1 representing roughly variance components of hybrid barley (Mühleisen et al. 2013), hybrid triticale (Mühleisen et al. 2015) and hybrid rice (J. Xu and X. Mi pers. comm).

We assumed a fixed total budget for producing DH lines and testcross seed, evaluating line per se performance in the observation nursery, genotyping and assessing testcross performance in field trials. The budget was expressed in field plot equivalents. A total budget available for hybrid breeding of 3000, 10,000 or 30,000 field plot equivalents was assumed (Table 1), reflecting the current basis of cereal breeding programs (Longin et al. 2014a; Cooper et al. 2014; Meng et al. 2016; A.E. Melchinger, H.P. Maurer, T. Miedaner pers. comm.). As an example the budget for the breeding strategy GSstandard is illustrated with:

$$B = N_{\text{ini}} ({\text{Cost}}_{\text{DH}} + {\text{Cost}}_{\text{nurserytest}} ) + N_{1} \left( {{\text{Cost}}_{\text{Genotyping}} } \right) + N_{2} T_{2} {\text{Cost}}_{\text{Hybridseed}} + N_{2} T_{2} L_{2} + N_{3} (T_{3} - T_{2} ){\text{Cost}}_{\text{Hybridseed}} + N_{3} T_{3} L_{3} ,$$

where CostDH, CostGenotyping and CostHybridseed refer to the cost of producing one DH line, genotyping one DH line and producing testcross seed of one DH line with one tester, respectively. Thus, the term \(N_{2} T_{2} {\text{Cost}}_{\text{Hybridseed}}\) represents the costs for hybrid seed production in test stage two, i.e., the costs for producing hybrids from N 2 DH lines crossed with T 2 testers. Similarly, the term \(N_{2} T_{2} L_{2}\) represents the costs for phenotyping these hybrids at L 2 locations. Based on discussions with different breeders and genotyping service providers, we assume the following scenarios for costs, each expressed as field plot equivalents: (1) CostDH = 1, CostGenotyping = 2, and CostHybridseed = 4 (Longin et al. 2014a, b; H.P. Maurer, pers. comm.), (2) CostDH = 1, CostGenotyping = 2, and CostHybridseed = 1, representing breeding programs with low cost of hybrid seed production (Virmani et al. 2003; Krchov and Bernardo 2015; A.E. Melchinger pers. comm.) and (3) CostDH = 3, CostGenotyping = 2, and CostHybridseed = 4, representing breeding programs with low availability of DH line production facilities (Table 1). We further assumed that one stage of hybrid seed production delivers enough seeds for 2 years of phenotyping. For instance, in two-stage phenotyping, the tester used in T 2 is also used in the following stage T 3. Thus, new testcross seed has to be produced only for the additional testers in stage three: \(N_{3} (T_{3} - T_{2} ){\text{Cost}}_{\text{Hybridseed}}\).

Before routinely using GS, calibration experiments are required to develop prediction models. Furthermore, recalibrations of models are regularly needed in the breeding program. We assumed, however, that GS calibrations were already developed outside the breeding cycle. Additionally, we assumed that recalibrations of the prediction models are based on the available data from the routine breeding, without additional budget requirements (Michel et al. 2016). According to current reports, prediction accuracy is on average 0.28 for wheat and maize in T0 scenarios (Zhao et al. 2013; Albrecht et al. 2014) and 0.36 for rice (Xu et al. 2014). We, therefore, assumed a prediction accuracy of 0.3, but additionally investigated prediction accuracies of 0.2 and 0.4.

Results

We investigated the effect of selection in observation nurseries on the expected annual selection gain of breeding strategies targeting grain yield. Traits assessed in the nurseries were assumed not to be correlated with grain yield, e.g., disease resistance and plant height in hybrid wheat, and we varied the selected fraction in the nursery (α) from 0.1 to 1. The breeding strategy GSrapid displayed the highest annual selection gain for grain yield independently of the fraction of lines selected in nurseries as well as the accuracy of genomic prediction (Fig. 2). Furthermore, neither the proportion of lines selected in nurseries nor the accuracy of GS affected the ranking of the evaluated breeding strategies, except for GSonly. For instance assuming a GS accuracy of 0.3, the ranking of the breeding strategies was GSrapid > PSrapid > GSstandard > PSstandard > GSonly. However, large effects of the accuracy of GS on the annual selection gain were observed for breeding strategy GSonly. Whereas this strategy was the poorest for a GS accuracy of 0.2, it ranked second best for a GS accuracy of 0.4 and α > 0.5. For all strategies using GS a higher selection intensity in the observation nursery, i.e., a smaller selected fraction α, led to reductions in the annual selection gain for grain yield for all three GS accuracies. This decrease in annual selection gain for grain yield was especially pronounced for α < 0.3.

Fig. 2
figure 2

Maximum annual selection gain for grain yield (ΔG a) of five hybrid breeding strategies according to the fraction of DH lines selected in the nursery and assuming an accuracy of genomic prediction of 0.2, 0.3 and 0.4 respectively. For details regarding breeding strategies see Fig. 1

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We next varied the ratio of variance components, the available budget, and the costs of DH and hybrid seed production and evaluated the annual selection gain always relative to the respective standard scenario. As described later, the selection in nurseries is an essential component in applied plant breeding programs and we, therefore, performed the following analyses for a selected fraction of α = 0.25. Variations of the variance components also affected the annual selection gain and the optimum allocation of test resources of the breeding strategies. The highest annual selection gain and the smallest standard deviation of selection gain were observed under variance component scenario 1, for all breeding strategies (Table 2). When \( \sigma_{{SCA}}^{2}\) was doubled (scenario 2), the annual selection gain decreased between 2.5 and 4.1 % depending on the breeding strategy. In addition, for the strategies GSrapid and PSrapid one additional tester in stage two was optimum compared to scenario 1. Reductions in annual selection gain ranging from 8.6 to 11.3 % were observed when doubling the non-genetic variance (scenario 3). In this scenario, the use of one additional tester in T 3 and two additional locations in L 2 were optimum for breeding strategies GSstandard and PSstandard. Despite the large variation of the variance components, the ranking of the breeding strategies was not affected with GSrapid having the highest annual selection gain.

Table 2 Optimum allocation of test resources maximizing selection gain (ΔG) under alternative hybrid breeding strategies for three variance components scenarios
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An increased budget led to a higher annual selection gain for all breeding strategies (Table 3). For instance, increasing the budget from 3000 to 30,000 field plot equivalents increased the annual selection gain of breeding strategy GSrapid by more than 43 %. Regardless of the amount of the budget, the ranking of the breeding strategies remained constant, i.e., GSrapid > PSrapid > GSstandard > PSstandard > GSonly. The optimum allocation of test resources was strongly affected by the amount of the budget, especially the number of test candidates, which increased strongly across all test stages with increasing budget.

Table 3 Optimum allocation of test resources maximizing selection gain (ΔG) under alternative hybrid breeding strategies for three different budgets of field plot equivalents
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Despite a large variation in the costs of DH line and hybrid seed production, the ranking of the breeding strategies remained unaffected, i.e., GSrapid > PSrapid > GSstandard > PSstandard > GSonly (Table 4). The annual selection gain in the breeding strategy GSrapid increased by 2 %, when the costs for hybrid seed production were reduced from 4 to 1 plot equivalent (scenario 2). Increasing the costs of DH line production (scenario 3) had large negative effects on all breeding strategies. For instance, the expected annual selection gain of strategy GSrapid was 7.5 % lower when the costs for DH line production increased from 1 to 3 plot equivalents. In all breeding strategies, the different costs affected mainly the number of test candidates in all selection stages, but the number of testers and locations remained fairly stable.

Table 4 Optimum allocation of test resources maximizing selection gain (ΔG) under alternative hybrid breeding strategies for three cost scenarios
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Discussion

A first study is available for hybrid wheat, where optimized breeding strategies including GS were compared based on their annual selection gain (Longin et al. 2015). The authors concluded that a breeding strategy with one stage GS followed by one stage phenotypic selection largely outperformed classical phenotypic selection. Besides restriction to wheat, limitations of the study were that only one set of variance components, one budget and fixed costs for DH line and hybrid seed production were assumed. Furthermore, the authors assumed that all produced DH lines enter grain yield testing. This assumption does, however, not reflect breeding reality, where new DH lines are first tested in small observation nurseries and selected for highly heritable traits like plant height, heading time and disease resistance (Cooper et al. 2014; Bassi et al. 2015). This motivated us to investigate the influence of nursery selection on the efficiency of hybrid breeding strategies targeting grain yield. For simplicity, we used as standard the scenario of hybrid wheat reported by Longin et al. (2015) and then separately varied the variance components, the budget and the costs of DH line and hybrid seed production. Finally, we discuss different scenarios that represent the situation of specific hybrid cereals, i.e., wheat, maize, barley, rye, rice and triticale, to determine the most efficient strategies for hybrid breeding including GS.

Before routinely using GS, calibration experiments are required to develop prediction models. Furthermore, recalibrations of GS models are regularly needed in the breeding program to warrant stable and high prediction accuracies. We assumed, however, that GS calibrations were already developed outside the breeding cycle in previous projects. Additionally, we assumed that recalibrations of the prediction models are based on the phenotypic and genotypic data routinely generated in the breeding program. Consequently, the breeder has not to invest extra money from his budget for these ameliorations of the GS prediction models (Michel et al. 2016).

Selection in nurseries does not affect the ranking of breeding strategies regarding annual selection gain for grain yield

Although we varied the fraction of lines selected in the observation nursery largely (α = 0.1 up to no selection for α = 1), the ranking of the evaluated breeding strategies based on the annual selection gain remained constant with GSrapid > PSrapid > GSstandard > PSstandard > GSonly (Fig. 2). The only exception was observed for GSonly when GS prediction accuracy was higher than 0.4. This constant ranking of the breeding strategies was especially surprising, as the optimum number of N ini was largely reduced in favor of an increased number of N 1 with decreasing selection intensity in nurseries, i.e., increasing α (data not shown). One explanation for the constant ranking might be that the main differences across the considered breeding strategies was a varying breeding cycle length with larger annual selection gain for shorter breeding strategies (Gordillo and Geiger 2008; Heffner et al. 2010).

The stricter the selection in the observation nursery the smaller was the annual selection gain for grain yield (Fig. 2). For instance for a GS accuracy of 0.3, GSrapid achieved 10.81 % less annual selection gain when comparing α = 0.1 with α = 1. Furthermore, with increasing GS prediction accuracy, the reduction of the annual selection gain for grain yield caused by strict nursery selection was even increased. These findings can be explained by two facts. First, nursery selection reduces the number of candidates tested for grain yield, thus reducing the selection intensity (cf. Becker 2011). Second, in our breeding strategies, GS on hybrid grain yield is directly following nursery selection based on line per se performance for traits not correlated with grain yield. The higher the GS prediction accuracy for grain yield is, the more worthy becomes a strategy to eliminate only the poorest performing lines in the observation nursery and to take more lines to a first prediction for hybrid grain yield—the trait of highest importance. However, high yielding hybrids with poor disease resistance, lodging or not adapted to specific growing conditions will never enter the market and consequently nursery selection on these traits is essential. Owing to the simplicity and the low costs to score these traits, it makes sense to screen these traits in a nursery before GS and grain yield phenotyping. This only moderately reduced the annual selection gain for grain yield, when the selected fraction of the nursery is not lower than α = 0.3 (Fig. 2). Based on many discussions with plant breeders and according to our results, we consequently used α = 0.25 for further analyses.

A smart re-allocation of the budget warrants highest selection gain

We investigated the impact of a wide range of (1) variance components, (2) budgets and (3) costs for DH line and testcross seed production on the ranking of breeding strategies and their optimum allocation of resources. Thereby, each single parameter was assessed under ceteris paribus conditions, i.e., varying only one parameter at a time and comparing the results with the standard scenario. Despite large differences in variance components, budgets and costs, the breeding strategy GSrapid always yielded the highest annual selection gain across all considered breeding strategies (Tables 2, 3, 4). This advantage of GSrapid can be explained by a reduction in the breeding cycle length and an increase of selection intensity compared to classical phenotypic selection (cf. Heffner et al. 2009; Jannink et al. 2010; Jonas and de Koning 2013; Heslot et al. 2015). For all investigated scenarios, GSrapid was 2 years faster and at least 38 % more DH lines could be investigated when compared to PSstandard (Tables 2, 3, 4). Consequently, we highly recommend a breeding strategy based on 1-year nursery observation followed by GS and a final 1-year phenotypic selection for hybrid cereal breeding. We concentrate on this strategy for the following discussion.

According to former reports, the implementation of GS in pig and maize breeding strategies was only beneficial in breeding programs with large budgets (Tribout et al. 2013; Riedelsheimer and Melchinger 2013). In contrast, we found a 32 % higher annual selection gain for the breeding strategy GSrapid compared to a classical phenotypic breeding strategy PSstandard for large but also small budgets (3000 or 30,000 field plots, Table 3). The difference to previous approaches might come from the slightly different underlying assumptions. While former studies concentrated only on one-stage selection in which training and prediction sets belonged to the same breeding cycle (Riedelsheimer and Melchinger 2013), we regarded multi-stage selection and assumed that prediction models were already established before starting a new breeding cycle. This approach has shown promising in maize and wheat (Zhao et al. 2013; Lehermeier et al. 2014; Michel et al. 2016). Consequently, we highly recommend the use of breeding strategy GSrapid also for breeding programs with small budgets.

Increasing the budget largely augmented the annual selection gain in all breeding strategies. However, the usefulness of a tenfold budget increase (from 3000 to 30,000 field plot equivalents) might be questionable, because only a 44 % increase in selection gain was obtained (Table 3). To make a more efficient use of the available budget, it might be smarter to change the breeding strategy. For instance, the annual selection gain of breeding strategy PSstandard with a budget of 10,000 plot equivalents was 0.79, whereas the annual selection gain of GSrapid with a budget of only 3000 plot equivalents was 0.85 (Table 3). In other words, this change of the breeding strategy realized a 7.6 % higher annual selection gain while requiring only 1/3 of the budget.

Previous studies on hybrid wheat breeding identified the costs of hybrid seed production as a main factor influencing annual selection gain (Longin et al. 2014a, b). However, for breeding strategies applying GS and nursery selection, the costs of DH production affected the annual selection gain more than the costs for hybrid seed production (Table 4). Using GS and nursery selection, the optimum number of initial DH lines N ini is largely increased, thus warranting a higher selection intensity but also higher costs for DH line production. In parallel, a considerably lower number of DH lines N 2 entering hybrid seed production and field testing was optimum for GSrapid compared to PSstandard explaining the reduced impact of decreasing hybrid seed production costs on the selection gain of GSrapid (Table 4). This highlights the need of efficient DH line production systems to fully exploit the potential of GS in hybrid cereal breeding programs. Summarizing, increasing the budget results in non-linear increases of the annual selection gain (cf. Longin et al. 2006; Becker 2011). It might thus be smarter to think about breeding strategy changes and invest the available budget in research for more efficient DH line technologies, reduced costs for marker data points and improved hybrid seed production systems.

Optimum allocation of test resources

For all scenarios tested in our study, the optimum allocation of test resources for GSrapid was found at the maximum possible number of locations (L 2 = 10). However, depending on the combination of budget, costs and variance components, T 2 varied between 2 and 5 (Tables 2, 3, 4). This intensive phenotyping is in agreement with former studies on the optimum allocation of resources in multi-stage selection (Utz 1969; Longin et al. 2007) and highlights the importance of high quality phenotyping when only one stage of field testing is available. The production of enough hybrid seeds for L 2 = 10 locations might be difficult for some hybrid breeding programs (Longin et al. 2014a; Bassi et al. 2015). We thus investigated alternatives minimizing the loss in annual selection gain. When a maximum number of L 2 = 5 was assumed, the number of testers T 2 should be increased from three to five minimizing the loss in annual selection gain to 2.7 % (data not shown). This shows the large flexibility in the allocation of resources around the optimum as already reported earlier (cf. Longin et al. 2006).

Previous studies in maize revealed an increase in the optimum number of test locations and testers when the proportion of non-genetic variance to GCA variance was increased (Longin et al. 2007). In our model calculations, only the number of testers T 2 used in the strategy GSrapid was increased from 3 to 5, when the non-genetic variance was doubled (Table 2). The increased number of testers was at the expense of a reduced number of candidates with 15 % less DH lines entering the nursery and GS stage and 32 % less candidates entering phenotypic selection on GCA.

Increasing the budget from 3000 to 30,000 field plot equivalents led only to minor changes in the optimum number of locations and testers. In contrast, the optimum number of test candidates was largely increased which is in line with previous findings (Robson et al. 1967; Knapp 1998; Longin et al. 2006; Krchov and Bernardo 2015). The costs of DH line production largely influenced the optimum number of candidates at the nursery and GS stage. For instance, for breeding strategy GSrapid, increasing DH production costs from 1 to 3 field plot equivalents nearly halved the number of DH lines entering the nursery (N ini) (Table 4). In contrast, the optimum number of candidates in the phenotyping stage N 2 remained constant. We speculate that this number N 2 is already at a minimum which should not be reduced anymore. Summarizing, changes in budget, variance components and costs for DH line and hybrid seed production largely affected the optimum allocation of test resources, underlining the necessity for breeders to regularly re-adjust their allocation of resources.

Breeding strategy GSrapid is best for hybrid breeding in wheat, maize, rye, barley, rice and triticale

Plant breeding programs are characterized by a unique combination of factors including but not limited to: biological and reproductive features of the crop, budget, costs and population specific parameters (reflected by variance components). With the advent of GS, each individual breeding program must decide if, how, and when to implement GS according to its specific resources and objectives (Jonas and de Koning 2013; Heslot et al. 2015; Bassi et al. 2015). While we described above the general effects of varying variance components, budgets and costs for DH line and hybrid seed production, we now want to relate these findings specifically to the cereal crops wheat, maize, rye, barley, rice and triticale.

We assumed that maize represents a well-established hybrid breeding crop with a large budget and low cost for DH line and hybrid seed production. We further assumed that early maturity maize could be bred realizing two growing seasons each year, resulting in a total breeding cycle length of three and 4 years for GSrapid and PSstandard, respectively. For late maturity maize, however, it might be difficult to realize hybrid seed production and field testing within 1 year. We, therefore, assumed that the breeding strategy is run until DH line nursery observation with two seasons per year but thereafter only with one season per year, extending its cycle length to four and 6 years for GSrapid and PSstandard, respectively. Hybrid wheat breeding programs have intermediate to high budgets with high costs for hybrid seed production (cf. Longin et al. 2012). The same holds true for barley with slightly smaller budgets (cf. Verstegen et al. 2014). For triticale, hybrid breeding looks promising (Oettler et al. 2005; Mühleisen et al. 2015; Boeven et al. 2016) but low budgets are coupled with high hybrid seed production costs. Further, we used the ratio of variance components of our study which were closest to reports on each crop in the literature. For all crops, we compared the annual selection gain achieved with the reference strategy PSstandard to the breeding strategy GSrapid (Table 5).

Table 5 Optimum allocation of test resources maximizing annual selection gain (ΔG a) in scenarios representing five different hybrid cereals
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GSrapid surpassed PSstandard by more than 24 % in all crops. The largest advantage was observed in maize (late maturity type) with 39.8 % followed by triticale with 35.7 %. Interestingly, triticale had the smallest budget considered in our study reinforcing our conclusion on the generally large advantage of implementing GS in breeding programs, also in those with small budget. When large budgets were available, i.e., for hybrid wheat and maize, the optimum number of DH lines entering nursery assessment was larger than 7500, which corresponds well to previous reports on resource allocation in applied maize breeding programs (Cooper et al. 2014). Given the high number of DH lines needed, current research efforts on DH line production, e.g., by improving haploid induction rate assessment, chromosome doubling techniques and phenotypic markers for the identification of haploid seeds (Chaikam et al. 2016; Melchinger et al. 2014, 2016a, b; Würschum et al. 2012, 2015) are of central importance to warrant the exploitation of GS in breeding of maize and other cereals.

The breeding strategies tested in this treatise are based on DH line production, but major biological and genetic features hamper practical implementation of DH lines for hybrid rye breeding (Geiger and Miedaner 2009). For instance, inbreeding depression with lethal effects and only small reductions in breeding cycle length have discouraged the use of DH lines in rye breeding (Tomerius 2001; Tenhola-Roininen et al. 2006). Thus, we assumed the use of S2 lines instead of DH lines for rye and used variance component scenario 1, which corresponds to current reports for variance components in rye breeding programs using S2 lines (Bernal-Vazquez et al. 2015). Furthermore, the breeding cycle length was increased by 1 year. Nevertheless, the use of GSrapid obtained a more than 24 % higher selection gain than the reference strategy PSstandard and is thus also very promising alternative to reduce breeding cycle length in hybrid rye.

For hybrid rice breeding, we assumed that two growing seasons per year can be run throughout the breeding strategies, resulting in a total breeding cycle length of three and 4 years for GSrapid and PSstandard, respectively. However, the number of available test locations is often limited to five or even less (J. Xu and X. Mi pers. comm) which might reduce the heritability and, thus, annual selection gain. Moreover, this reduction might have a stronger effect on strategies with only one stage of phenotypic selection as GSrapid, compared to two-stage phenotypic selection as PSstandard. Therefore, we developed additional simulations varying the number of locations L j only between 1 and 5 for rice. We found a 28.5 % higher annual selection gain for GSrapid compared with PSstandard. Additionally, the limitation to a maximum number of five test locations resulted in an increased optimum number of testers. In conclusion, we highly recommend the use of GS coupled with moderately high selection in nurseries for hybrid breeding in cereals.

Hybrid seed production works well in maize with isolation fields and in wheat with the chemical hibridization agent (CHA) Crosoir 100® (Saaten-Union Recherche). For simplicity of the investigated breeding strategies, we assumed that a CHA also works well in barley, tritcale, rye and rice. However, the use of a cytoplasmic male sterility (CMS) system is common practice for rice, barley, rye, and triticale (Longin et al. 2012) and under development for wheat. Hopefully, the restorer gene(s) will be rapidly fixed in the male pool and, thus, the breeding strategies shown in this treatise can directly be applied. The optimum breeding strategy of the female pool based on a CMS system, however, requires further research. In addition, breeders commonly select on multiple traits in parallel. Thus, an investigation of index selection in the calculation of selection gain and its impact on various breeding strategies and their optimum allocation of resources is of high interest but yet lacking in the literature.

Author contribution statement

C. F. H. Longin designed the study. J. J. Marulanda wrote scripts for simulations, expanded the R package for optimizing breeding strategies including nursery selection and performed data analysis. X. Mi assisted in modifications of the R package. J. J. Marulanda, T. Würschum, X. Mi, A. E. Melchinger, J. Xu and C. F. H. Longin wrote the manuscript. All authors read and reviewed the manuscript.