Artificial Neural Networks for Modeling and Optimizing Egg Cost in Second-Cycle Laying Hens Based on Dietary Intakes of Essential Amino Acids

Abstract

Egg production is a significant source of animal protein for human consumption. Feed costs significantly impact the profitability of egg production, representing more than 70% of the variable costs. This study evaluated the effect of dietary intakes of three essential amino acids (EAAs) on the egg cost for H&N Brown second-cycle laying hens. The hens were fed for 20 weeks with 23 diets that varied in their lysine, methionine + cystine, and threonine contents. These amino acids were derived from both dietary and synthetic sources. Zootechnical results were used to calculate the feed cost per kilogram of egg (FCK), considering the cost of raw materials and the diet composition. Multivariate polynomial models and artificial neural networks (ANNs) were validated to predict FCK as a function of the EAAs and time. The EAA intakes that minimize FCK over time were optimized using the best model, a cascade-forward ANN with a softmax transfer function. The optimal scenario for FCK (0.873 USD/kg egg) at 20 weeks was achieved at 943.7 mg lysine/hen-day, 858.3 mg methionine + cystine/hen-day, and 876.8 mg threonine/hen-day. ANNs could be a valuable tool for predicting the egg cost of laying hens based on the nutritional requirements. This could help improve economic efficiency and reduce the feed costs in poultry companies.

1. Introduction

According to FAO estimations, the world population may increase to 9.8 billion by 2050 [1]. This accelerated growth has aroused worldwide concern, mainly due to the difficulty of producing more food without affecting food security and of making it accessible to the entire population. Among the possible strategies that can be implemented to increase food production worldwide, egg production is emerging as a viable alternative to advance progress toward the second Sustainable Development Goal (end hunger, achieve food security and improved nutrition, and promote sustainable agriculture) [2].

From a nutritional point of view, eggs are an essential source of proteins, phospholipids, vitamins (e.g., thiamin, riboflavin, niacin, vitamin B12, vitamin A, vitamin E, vitamin D, and vitamin K), minerals (e.g., iron, selenium, calcium, magnesium, phosphorus, potassium, sodium, and zinc) and antioxidants, and their consumption could cover part of the daily needs of these types of nutrients [3,4]. Furthermore, this product presents advantages over other protein sources, primarily stemming from its low commercial cost, which has increased its consumption and production worldwide. It is estimated that between 2018 and 2020, egg production increased by more than 8% worldwide [5]. Despite the growth of this industry and the egg’s nutritional value, some factors limit its production, such as costs associated with animal feed, which can exceed 70% of variable costs [6]. An effective strategy to reduce egg production costs is the implementation of second production cycles in birds. By adopting this method, different benefits are obtained, such as increased laying cycles and egg size, which provide responsiveness to fluctuations in costs associated with production supplies [7]. For second-cycle production to be efficient, the nutritional requirements of laying birds must be defined. However, these requirements vary according to genetic lines, production volume, environmental conditions, and the age of the birds [8].

Amino acids are critical components in the nutrition and productivity of laying hens. Therefore, a deficit or excess of these compounds can be configured as a disadvantage from an economic point of view [9]. Among the amino acids that are part of the diet of laying hens, lysine (Lys), sulfur amino acids (methionine and cysteine; Met + Cys), and threonine (Thr) are the most relevant. Lys is an essential amino acid (EAA) for the maintenance, growth, and production of birds, with the primary function of participating in body protein synthesis [9,10]. As plant-based protein sources have low levels of Met + Cys, these are the first limiting amino acids for poultry diets [9]. The third limiting amino acid in poultry diets is Thr [11], which is related to maintaining the integrity of the intestinal barrier and the production of antibodies that play an essential role in bird immunity [9]. Several studies have evaluated the optimal ratio of amino acids in the diet of laying hens to improve productivity or egg size [12,13]. However, only some studies have focused on minimizing the feed cost per kilogram of egg (FCK) by evaluating the nutritional requirements associated with amino acids in the diet.

The challenges and trends in poultry require investment in technology and innovation, especially in data analysis, to improve the industry’s efficiency, profitability, sustainability, and competitiveness [14]. Data analysis can support planning and cost reduction in egg production by estimating the productive behavior of laying hens under various nutritional, sanitary, environmental, and economic conditions [10,12,15,16].

Predictive models are valuable tools to address the challenges of poultry production in terms of business management and cost reduction [17,18,19]. They can provide benefits such as enhancing the control and monitoring of farm indicators to identify health or nutrition issues; conducting comparative analysis between different production conditions to assess the performance and profitability of companies and identify areas for improvement or correction; and supporting companies to adapt to market or environmental changes, estimate scenarios, and plan actions aligned with their objectives.

In this study, two models based on ANNs were applied to assess the dependence of the feed cost per kilogram of egg (FCK) on the dietary intakes of the three main EAAs for laying hens. The model with the best fit was used to minimize FCK and to determine the relative importance of Lys, Met + Cys, and Thr in H&N Brown second-cycle laying hens (SCLHs) under field conditions.

2. Materials and Methods

2.1. Fieldwork, Experimental Design, and Diet Formulation

This study was conducted on a commercial farm in San Pedro, Antioquia (Colombia). The university bioethics committee approved the protocols and experimental procedures on the laying hens (Act 03/2017). For the fieldwork, H&N Brown SCLHs aged 91 weeks were used (1380 hens). The animals were housed in California-type cages under environmental conditions and distributed in a layer shed after being subjected to proper molting and resting protocols [12]. In each cage, 12 hens were placed (576 cm²/bird), and each cage was fitted with a PVC side feeder and two nipple drinkers. Five cages were used as replicates for each treatment (60 birds per treatment).

The diet formulation per treatment and the feed consumption were used to express the design matrix as dietary intakes of Lys (iLys), Met + Cys (iMetCys), and Thr (iThr) (

Table 1. Essential amino acid intakes of H&N Brown SCLHs over 20 weeks and diet cost.

Diet	iLys * (mg/hen-day)	iMetCys * (mg/hen-day)	iThr * (mg/hen-day)	Diet Cost (USD/kg)
1	727.0 ± 10.0	661.9 ± 9.1	552.1 ± 7.6	0.3974
2	732.4 ± 9.6	666.8 ± 8.8	875.2 ± 11.5	0.4015
3	728.4 ± 18.0	852.1 ± 21.0	712.4 ± 17.6	0.4036
4	728.5 ± 17.2	1042.3 ± 24.6	553.2 ± 13.0	0.4058
5	736.3 ± 18.6	1053.5 ± 26.7	879.8 ± 22.3	0.4099
6	809.6 ± 11.3	736.6 ± 10.3	614.6 ± 8.6	0.4015
7	812.7 ± 18.0	739.5 ± 16.4	805.8 ± 17.9	0.4040
8	828.4 ± 22.7	982.6 ± 26.9	628.9 ± 17.2	0.4065
9	810.2 ± 11.3	961.9 ± 13.4	803.3 ± 11.2	0.4089
10	951.7 ± 17.8	673.7 ± 12.6	723.7 ± 13.6	0.4033
11	947.4 ± 19.0	861.7 ± 17.3	559.4 ± 11.2	0.4055
12	938.4 ± 13.6	853.6 ± 12.4	713.6 ± 10.3	0.4075
13	943.7 ± 12.2	858.3 ± 11.1	876.8 ± 11.3	0.4095
14	935.1 ± 12.9	1040.2 ± 14.3	711.0 ± 9.8	0.4117
15	1076.9 ± 23.4	751.2 ± 16.3	626.8 ± 13.6	0.4061
16	1066.3 ± 15.7	743.9 ± 10.9	810.7 ± 11.9	0.4085
17	1058.8 ± 14.6	963.9 ± 19.5	616.3 ± 12.5	0.4111
18	1058.5 ± 14.6	963.6 ± 13.3	804.7 ± 11.1	0.4135
19	1164.7 ± 30.3	674.3 ± 17.7	562.5 ± 14.7	0.4052
20	1143.1 ± 15.8	661.9 ± 9.1	868.8 ± 12.0	0.4092
21	1154.2 ± 18.4	858.7 ± 13.7	771.9 ± 11.4	0.4114
22	1153.3 ± 20.7	1049.5 ± 18.8	557.0 ± 10.0	0.4135
23	1159.5 ± 16.9	1055.1 ± 15.4	881.2 ± 12.3	0.4176

* Values expressed as means ± standard deviations (n = 100). Source: Morales-Suárez et al. [12].

) according to the methodology and results of Morales-Suárez et al. [12]. This methodology combined levels of the three EAAs from a comprehensive literature review using a central composite design. The resulting 23 diets (treatments) were prepared weekly by adding L-lysine (98.5% feed grade), DL-methionine (99% feed grade), and L-threonine (98.5% feed grade) to a diet based on corn and soybean meal (Supplementary Table S1). Thus, the Lys, Met + Cys, and Thr levels in the experimental diets were adjusted by considering the contributions of the synthetic amino acids and food ingredients. The layers were fed at 115 g/hen-day for 20 weeks.

In this way, the experiment considered the dependence of FCK on four factors: iLys, iMetCys, iThr, and time.

2.2. Zootechnical and Economic Results

The feed conversion ratio (CR) was computed weekly as the ratio between kilograms of feed intake and kilograms of eggs.

The feed cost per kilogram of egg (FCK) was calculated considering the cost of the diets (Table 1) calculated from the local cost of the raw materials in USD in March 2021 (

Table 2. Cost of diet ingredients.

Ingredient	Cost (USD/kg)
Corn	0.29
Soybean meal	0.53
Palm oil	0.93
Corn gluten	0.92
Calcium carbonate	0.05
Monocalcium phosphate	0.62
Bentonite	0.06
Choline chloride	1.15
Salt	0.08
Vitamin premix	11.78
Mineral premix	6.42
L-lysine *	1.73
DL-Methionine *	2.56
L-threonine *	1.50
Tryptophan	9.96
Arginine	15.37
Sodium bicarbonate	0.44
Valine	4.20

* Synthetic amino acids used as dietary supplements.

). For each treatment, the FCK in USD/kg egg was estimated weekly by multiplying the cost per kilogram of feed and CR.

2.3. Mathematical Modeling

The FCK was modeled as a function of dietary intake of EEAs (iLys, iMetCys, iThr) and time (t) using multivariable polynomial models and ANNs.

The second- (Equation (1)) and third-order (Equation (2)) polynomial models were evaluated by means of stepwise regression to include only statistically significant terms at a 95% confidence level [20].

$$\mathrm{F}\mathrm{C}\mathrm{K}={\mathsf{\beta }}_0+\sum _{\mathrm{j}=1}^4{\mathsf{\beta }}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}+\sum _{\mathrm{j}=1}^3\sum _{\mathrm{k}>\mathrm{j}}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{k}}{\mathrm{x}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{k}}+\sum _{\mathrm{j}=1}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}^2$$

(1)

$$\mathrm{F}\mathrm{C}\mathrm{K}={\mathsf{\beta }}_0+\sum _{\mathrm{j}=1}^4{\mathsf{\beta }}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}+\sum _{\mathrm{j}=1}^3\sum _{\mathrm{k}>\mathrm{j}}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{k}}{\mathrm{x}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{k}}+\sum _{\mathrm{j}=1}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}^2+\sum _{\mathrm{j}=1}^2\sum _{\mathrm{k}>\mathrm{j}}^3\sum _{\mathrm{l}>\mathrm{k}}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{k}\mathrm{l}}{\mathrm{x}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{k}}{\mathrm{x}}_{\mathrm{l}}\sum _{\mathrm{j}=1}^4\sum _{\mathrm{k}\ne \mathrm{j}}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{k}\mathrm{k}}{\mathrm{x}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{k}}^2+\sum _{\mathrm{j}=1}^4{\mathsf{\beta }}_{\mathrm{j}\mathrm{j}\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}^3$$

(2)

Here, FCK is the feed cost per kilogram of egg (USD/kg egg); x terms represent the EEA dietary intakes (iLys, iMetCys, iThr) and the time; and β terms are the parameters associated with linear, quadratic, cubic, and cross-product terms.

Additionally, feed-forward (Equation (3)) (Figure 1a) and cascade-forward (Equation (4)) (Figure 1b) ANN architectures with one hidden layer were used to evaluate the dependence of the FCK on the independent factors. These network architectures have been used due to their prediction capability for complex multivariate nonlinear problems and their modeling capability for diverse natural phenomena that are very difficult to handle via classical techniques [21]. Five to eleven neurons and four transfer functions (hyperbolic tangent, log sigmoid, radial basis, and softmax) (Equations (5)–(8), respectively) were assessed for each ANN architecture [22].

$$\mathrm{F}\mathrm{C}\mathrm{K}={\mathrm{w}}_{\mathrm{h}\mathrm{o}}\times \mathrm{f}+{\mathrm{b}}_{\mathrm{o}}$$

(3)

$$\mathrm{F}\mathrm{C}\mathrm{K}={\mathrm{w}}_{\mathrm{h}\mathrm{o}}\times \mathrm{f}+{\mathrm{w}}_{\mathrm{i}\mathrm{o}}\times \mathrm{x}+{\mathrm{b}}_{\mathrm{o}}$$

(4)

$$\mathrm{f}=\frac{2}{1+{\mathrm{e}}^{\left[-2\left({\mathrm{w}}_{\mathrm{i}\mathrm{h}} \times \mathrm{x} + {\mathrm{b}}_{\mathrm{h}}\right)\right]}}-1$$

(5)

$$\mathrm{f}=\frac{1}{1+{\mathrm{e}}^{\left[-\left({\mathrm{w}}_{\mathrm{i}\mathrm{h}} \times \mathrm{x} + {\mathrm{b}}_{\mathrm{h}}\right)\right]}}$$

(6)

$$\mathrm{f}={\mathrm{e}}^{\left[-{\left({\mathrm{w}}_{\mathrm{i}\mathrm{h}} \times \mathrm{x} + {\mathrm{b}}_{\mathrm{h}}\right)}^2\right]}$$

(7)

$$\mathrm{f}=\frac{{\mathrm{e}}^{\left({\mathrm{w}}_{\mathrm{i}\mathrm{h}} \times \mathrm{x} + {\mathrm{b}}_{\mathrm{h}}\right)}}{\sum {\mathrm{e}}^{\left({\mathrm{w}}_{\mathrm{i}\mathrm{h}} \times \mathrm{x} + {\mathrm{b}}_{\mathrm{h}}\right)}}$$

(8)

Here, FCK is the output estimation, w_ho is the weights between the hidden and the output layers, f is the transfer function, w_io is the weights between the input and the output layers in the cascade-forward architecture, x is the input predictor (iLys, iMetCys, iThr, t), b_o is the bias of the output layer, w_ih is the weights between the input and the hidden layers, and b_h is the biases of the hidden layer.

2.4. Training, Validation, and Statistical Analysis

MATLAB^® R2019a (The MathWorks Inc., Natick, MA, USA) was used for the analysis of two sets of data: the training group with 1380 data records (three treatment replicates) and the validation group with 920 data records (two treatment replicates).

The statistically significant parameters of the multivariate polynomial models (Equations (1) and (2)) were obtained through the MATLAB^® function “stepwisefit”. The parameters of the ANN models (biases and weights) were identified using the “trainbr” function of MATLAB^®, which avoids overfitting in the training process through a Bayesian regularization algorithm [23].

The accuracy, goodness of fit, and deviation of the models were evaluated by examining the root-mean-square error (RMSE) (Equation (9)), the adjusted coefficient of determination (R²_adj) (Equation (10)), and the bias (Equation (11)), respectively. Meanwhile, the quality of the models was determined by employing the Akaike Information Criterion (AIC) (Equation (12)) [24]. The models with the best fit were evaluated for the adequacy of the estimates using a residual analysis.

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{F}\mathrm{C}\mathrm{K}}_{\mathrm{i}}^{\mathrm{*}}-{\mathrm{F}\mathrm{C}\mathrm{K}}_{\mathrm{i}}\right)}^2}{\mathrm{n}}}$$

(9)

$${\mathrm{R}}_{\mathrm{a}\mathrm{d}\mathrm{j}}^2=1-{\mathrm{S}}_{\mathrm{y}\mathrm{x}}^2/{\mathrm{S}}_{\mathrm{y}}^2$$

(10)

$$\mathrm{b}\mathrm{i}\mathrm{a}\mathrm{s}=\frac{1}{\mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{F}\mathrm{C}\mathrm{K}}_{\mathrm{i}}^{\mathrm{*}}-{\mathrm{F}\mathrm{C}\mathrm{K}}_{\mathrm{i}}\right)$$

(11)

$$\mathrm{A}\mathrm{I}\mathrm{C}=\mathrm{n}\log \left[\mathrm{d}\mathrm{e}\mathrm{t}\left(\frac{1}{\mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathsf{\epsilon }{\mathsf{\epsilon }}^{\mathrm{T}}\right)\right]+2{\mathrm{n}}_{\mathrm{p}}+\mathrm{n}\left[\log \left(2\mathsf{\pi }\right)+1\right]$$

(12)

Here, ${\mathrm{F}\mathrm{C}\mathrm{K}}_{\mathrm{i}}^{\mathrm{*}}$ and ${\mathrm{F}\mathrm{C}\mathrm{K}}_{\mathrm{i}}$ represent the observed and predicted values, respectively; S_y and S_yx are the standard deviations of the sample and model estimations, respectively; ε is the prediction error; and n and n_p are the numbers of data points and estimated parameters, respectively. A high value of R²_adj and low values of RMSE and AIC were used to select the best model.

2.5. Optimization

An optimization problem was formulated to find the iLys, iMetCys, and iThr values that minimize FCK (Equation (13)) at 4, 8, 12, 16, and 20 weeks. The optimization was solved for each of the weeks mentioned above using the MATLAB^® function “fmincon” and constraining the limits to the maximum and minimum values of iLys, iMetCys, and iThr used in the experiment (Table 1).

$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} \mathrm{F}\mathrm{C}\mathrm{K}\left(\mathrm{i}\mathrm{L}\mathrm{y}\mathrm{s}, \mathrm{i}\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{C}\mathrm{y}\mathrm{s}, \mathrm{i}\mathrm{T}\mathrm{h}\mathrm{r}\right)\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\left\{\begin{array}{c}{\mathrm{i}\mathrm{L}\mathrm{y}\mathrm{s}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{i}\mathrm{L}\mathrm{y}\mathrm{s}\le {\mathrm{i}\mathrm{L}\mathrm{y}\mathrm{s}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\\ {\mathrm{i}\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{C}\mathrm{y}\mathrm{s}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{i}\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{C}\mathrm{y}\mathrm{s}\le {\mathrm{i}\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{C}\mathrm{y}\mathrm{s}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\\ {\mathrm{i}\mathrm{T}\mathrm{h}\mathrm{r}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{T}\mathrm{h}\mathrm{r}\le {\mathrm{i}\mathrm{T}\mathrm{h}\mathrm{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}\right.$$

(13)

3. Result

3.1. Egg Cost and Zootechnical Results

The observed FCK values were between 0.998 and 1.559 USD/kg egg in week 8, between 0.949 and 1.214 USD/kg egg in week 12, between 0.928 and 1.090 USD/kg egg in week 16, and between 0.919 and 1.041 USD/kg egg in week 20 (

Table 3. Feed cost per kilogram of egg (USD/kg egg) of H&N Brown SCLHs for the diets over 20 weeks.

Diet	Week 8	Week 12	Week 16	Week 20
1	1.33 ± 0.04	1.15 ± 0.06	1.06 ± 0.06	1.02 ± 0.05
2	1.56 ± 0.03	1.21 ± 0.03	1.09 ± 0.03	1.04 ± 0.04
3	1.16 ± 0.02	1.02 ± 0.03	0.96 ± 0.03	0.94 ± 0.03
4	1.15 ± 0.01	1.03 ± 0.02	0.99 ± 0.02	0.96 ± 0.03
5	1.18 ± 0.02	1.05 ± 0.03	0.98 ± 0.02	0.95 ± 0.02
6	1.43 ± 0.02	1.16 ± 0.03	1.06 ± 0.03	1.02 ± 0.03
7	1.23 ± 0.04	1.06 ± 0.04	0.98 ± 0.04	0.94 ± 0.03
8	1.41 ± 0.03	1.15 ± 0.04	1.04 ± 0.04	0.99 ± 0.04
9	1.21 ± 0.01	1.07 ± 0.02	1.01 ± 0.03	0.98 ± 0.03
10	1.22 ± 0.03	1.08 ± 0.04	1.00 ± 0.03	1.01 ± 0.09
11	1.16 ± 0.02	1.06 ± 0.03	0.99 ± 0.02	0.97 ± 0.02
12	1.12 ± 0.05	1.01 ± 0.02	0.96 ± 0.02	0.94 ± 0.02
13	1.00 ± 0.04	0.95 ± 0.02	0.93 ± 0.04	0.92 ± 0.04
14	1.17 ± 0.02	1.05 ± 0.01	1.00 ± 0.02	0.97 ± 0.03
15	1.09 ± 0.03	1.02 ± 0.03	0.97 ± 0.03	0.95 ± 0.03
16	1.25 ± 0.01	1.11 ± 0.01	1.03 ± 0.02	1.00 ± 0.02
17	1.21 ± 0.02	1.04 ± 0.02	0.98 ± 0.02	0.95 ± 0.03
18	1.28 ± 0.05	1.09 ± 0.03	1.03 ± 0.03	1.00 ± 0.03
19	1.36 ± 0.04	1.14 ± 0.01	1.05 ± 0.02	1.01 ± 0.01
20	1.34 ± 0.02	1.16 ± 0.03	1.06 ± 0.04	1.02 ± 0.04
21	1.35 ± 0.03	1.14 ± 0.04	1.05 ± 0.04	1.02 ± 0.04
22	1.48 ± 0.03	1.20 ± 0.03	1.07 ± 0.03	1.03 ± 0.04
23	1.41 ± 0.04	1.21 ± 0.02	1.09 ± 0.02	1.04 ± 0.02

Values expressed as means ± standard deviations (n = 5).

). FCK generally decreases over time since feed costs are amortized as the hens’ egg production increases over the production cycle.

The lowest FCK corresponds to diet 13 at weeks 8, 12, 16, and 20, followed by diet 12 at weeks 12, 16, and 20 and diet 3 at week 20 (Table 3). In these diets, the iLys values ranged from 728.4 mg/hen-day (diet 3) to 943.7 mg/hen-day (diet 13); the iMetCys values ranged between 852.1 mg/hen-day (diet 3) and 858.3 mg/hen-day (diet 13); and the iThr values ranged between 712.4 mg/hen-day (diet 3) and 876.8 mg/hen-day (diet 13) (Table 1).

The observed CR was between 2.02 and 2.36 kg feed/kg egg, comparable to the values for commercial hens at 20 weeks of production [25]. Diet 13 presented CR values between 3.40 ± 0.19 kg feed/kg egg (week 4) and 2.18 ± 0.16 kg feed/kg egg (week 20) (

Table 4. Feed conversion ratio (kg feed/kg egg) of H&N Brown SCLHs for the diets over 20 weeks.

Diet	Week 8	Week 12	Week 16	Week 20
1	3.43 ± 0.23	2.58 ± 0.12	2.16 ± 0.33	2.22 ± 0.06
2	3.62 ± 0.20	2.62 ± 0.14	2.03 ± 0.15	2.03 ± 0.11
3	2.89 ± 0.11	2.37 ± 0.11	2.08 ± 0.25	2.25 ± 0.17
4	2.88 ± 0.18	2.20 ± 0.14	2.09 ± 0.26	2.18 ± 0.24
5	3.45 ± 0.44	1.88 ± 0.08	2.15 ± 0.28	1.95 ± 0.14
6	5.05 ± 0.53	2.17 ± 0.07	2.00 ± 0.23	2.03 ± 0.10
7	3.50 ± 0.36	2.00 ± 0.11	1.99 ± 0.30	2.09 ± 0.22
8	4.20 ± 0.29	2.16 ± 0.09	1.99 ± 0.18	2.08 ± 0.10
9	2.87 ± 0.26	2.30 ± 0.15	1.89 ± 0.07	2.17 ± 0.23
10	3.25 ± 0.30	2.34 ± 0.17	2.27 ± 0.23	2.09 ± 0.17
11	3.59 ± 0.26	2.10 ± 0.04	2.11 ± 0.19	2.23 ± 0.19
12	3.05 ± 0.30	2.01 ± 0.05	2.03 ± 0.18	2.14 ± 0.23
13	2.48 ± 0.20	2.08 ± 0.16	2.11 ± 0.21	2.18 ± 0.27
14	3.40 ± 0.19	2.12 ± 0.05	2.05 ± 0.14	2.17 ± 0.11
15	2.59 ± 0.30	2.26 ± 0.16	2.14 ± 0.22	2.59 ± 1.33
16	3.10 ± 0.10	2.43 ± 0.07	2.19 ± 0.21	2.24 ± 0.23
17	3.30 ± 0.22	2.25 ± 0.03	2.00 ± 0.05	2.18 ± 0.14
18	3.50 ± 0.13	2.15 ± 0.09	2.03 ± 0.14	2.29 ± 0.18
19	3.84 ± 0.30	2.35 ± 0.11	2.18 ± 0.08	2.05 ± 0.15
20	4.07 ± 0.53	2.36 ± 0.14	2.16 ± 0.10	2.18 ± 0.19
21	3.48 ± 0.21	2.34 ± 0.17	2.21 ± 0.19	2.08 ± 0.08
22	4.39 ± 0.16	2.35 ± 0.14	2.14 ± 0.26	2.05 ± 0.08
23	3.60 ± 0.18	2.39 ± 0.15	2.11 ± 0.04	2.02 ± 0.10

Values expressed as means ± standard deviations (n = 5).

).

Egg production ranged from 83.1% to 85.3%, which agrees with the values reported for commercial hens at 20 weeks of production in the Hy-Line Brown management guide [25]. Diet 13, the treatment with the lowest FCK, presented the following egg production rates: 72.1 ± 4.7% (week 4); 80.9 ± 5.1% (week 8); 81.2 ± 9.1% (week 12); 77.1 ± 6.5% (week 16); and 80 ± 4.9% (week 20) [12]. At week 20, diet 13 showed 105.1 ± 4.2 hen-housed eggs, an average of 5.25 eggs per week.

3.2. Curve Fitting and Statistical Criteria

FCK was modeled using two different approaches (

Table 5. Modeling results for the feed cost per kilogram of egg (FCK) of H&N Brown SCLHs.

Model	Specifications	Training		Validation		p-Value	bias	AIC
		RMSE	R2adj	RMSE	R2adj
Multivariate polynomial
Second order	12 parameters	0.108	0.7663	0.111	0.7485	0.001	2.38 × 10−5	19028
Third order	28 parameters	0.082	0.8642	0.084	0.8519	0.001	7.11 × 10−5	18621
ANN
Feed-forward network	Softmax transfer function Nine hidden neurons 55 parameters	0.047	0.9418	0.045	0.9537	0.001	2.38 × 10−7	17714
Cascade-forward network	Softmax transfer function Nine hidden neurons 59 parameters	0.042	0.9534	0.043	0.9564	0.001	1.23 × 10−7	17608

RMSE is the root-mean-square error; R²_adj is the adjusted coefficient of determination; AIC is Akaike’s information criterion; p-value is the probability to test the validity of the null hypothesis that the residuals came from a normal distribution, according to the Lilliefors test.

). In the first approach, the multivariate polynomial models presented RMSE values of 0.0818 and 0.0841 and R²_adj values of 0.8642 and 0.8519 for the training and validation, respectively.

The best polynomial model for FCK was the third-order equation in Equation (14), where the independent variables iLys, iMetCys, iThr, and t are represented as x₁, x₂, x₃, and x₄, respectively.

$$\begin{array}{c}\mathrm{F}\mathrm{C}\mathrm{K}=-3.0975+0.1523{\mathrm{x}}_1-0.1945{\mathrm{x}}_2+7.009\times {10}^{-2}{\mathrm{x}}_3-0.6893{\mathrm{x}}_4+5.901\times {10}^{-5}{\mathrm{x}}_1{\mathrm{x}}_2+1.653\times {10}^{-4}{\mathrm{x}}_2{\mathrm{x}}_3\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}-2.323\times {10}^{-4}{\mathrm{x}}_1{\mathrm{x}}_3+6.095\times {10}^{-4}{\mathrm{x}}_4{\mathrm{x}}_1+4.990\times {10}^{-4}{\mathrm{x}}_4{\mathrm{x}}_2-3.589\times {10}^{-4}{\mathrm{x}}_4{\mathrm{x}}_3-1.073\times {10}^{-4}{\mathrm{x}}_1^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+1.192\times {10}^{-4}{\mathrm{x}}_2^2-4.103\times {10}^{-5}{\mathrm{x}}_3^2+2.060\times {10}^{-2}{\mathrm{x}}_4^2+9.065\times {10}^{-8}{\mathrm{x}}_1^2{\mathrm{x}}_3-2.760\times {10}^{-7}{\mathrm{x}}_1^2{\mathrm{x}}_4\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}-3.276{\times {10}^{-8}\mathrm{x}}_2^2{\mathrm{x}}_1-9.762{\times {10}^{-8}\mathrm{x}}_2^2{\mathrm{x}}_3-2.298{\times {10}^{-7}\mathrm{x}}_2^2{\mathrm{x}}_4+4.207{\times {10}^{-8}\mathrm{x}}_3^2{\mathrm{x}}_1\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+1.174{\times {10}^{-7}\mathrm{x}}_3^2{\mathrm{x}}_4-1.952{\times {10}^{-7}\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_4+1.131{\times {10}^{-7}\mathrm{x}}_2{\mathrm{x}}_3{\mathrm{x}}_4+{9.898\times {10}^{-8}\mathrm{x}}_1{\mathrm{x}}_3{\mathrm{x}}_4\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+1.710\times {10}^{-8}{\mathrm{x}}_1^3-5.055\times {10}^{-9}{\mathrm{x}}_2^3-4.380\times {10}^{-4}{\mathrm{x}}_4^3\hfill \end{array}$$

(14)

The best results for FCK using ANN models were obtained using a cascade-forward architecture, nine neurons in the hidden layer, and a softmax transfer function (Table 5). The fitting results presented RMSE values of 0.042 and 0.043 and R²_adj values of 0.9534 and 0.9564 for the training and validation, respectively.

The ANN models showed better-fitting results than other modeling approaches, presenting higher R²_adj, lower RMSE, and lower AIC values. In addition, the ANN models predicted the behavior of FCK using a simple network architecture (Equations (4) and (7)), which would facilitate the subsequent analysis and optimization. The parameters (weights and biases) of the best ANN model are presented in

Table 6. Weights and biases of the ANN model for feed cost per kilogram of egg (FCK) of H&N Brown SCLHs.

wih	who	wio	bh	bo
$\left[\begin{array}{rrrr}1.5971& 0.8748& -2.4899& 1.1157\\ 19.078& 4.7011& 1.5910& -3.5728\\ 18.684& 4.4364& 1.7995& 0.4581\\ 4.4308& -4.2488& -0.8286& 0.1034\\ -17.374& 1.1351& 3.4314& -2.8917\\ 0.5897& 0.7214& -2.6382& 1.6316\\ -10.577& -11.288& 1.3771& 1.9248\\ -15.701& 2.4790& 0.9823& 2.3833\\ -0.7277& 1.1889& -3.2246& -1.1524\end{array}\right]$	${\left[\begin{array}{r}-0.00138\\ 0.00223\\ -0.00131\\ 0.00121\\ -0.00079\\ 0.00090\\ -0.00114\\ -0.00127\\ 0.00265\end{array}\right]}^{\mathrm{T}}$	${\left[\begin{array}{r}5.35\times {10}^{-5}\\ 1.61\times {10}^{-5}\\ -1.56\times {10}^{-6}\\ -3.69\times {10}^{-5}\end{array}\right]}^{\mathrm{T}}$	$\left[\begin{array}{r}8.9212\\ -3.6435\\ 3.2475\\ 0.1961\\ -8.2080\\ 5.1132\\ -8.5495\\ 0.1083\\ 2.8147\end{array}\right]$	$0.00111$

w_ih is the weights between the input and the hidden layers; w_ho is the weights between the hidden and the output layers; b_h is the biases of the hidden layer; b_o is the bias of the output layer.

.

3.3. Optimization

The minimum FCK during 20 weeks of production was obtained with mean amino acid levels of 909.5 mg/hen-day of iLys, 830.71 mg/hen-day of iMetCys, and 881 mg/hen-day of iThr (

Table 7. Optimal intakes of lysine, methionine + cysteine, and threonine between 8 and 20 weeks of production for H&N Brown SCLHs considering the feed cost per kilogram of egg (FCK) as the optimization criterion.

Week	Minimum FCK (USD/kg egg)	Optimal Intakes (mg/hen-day)
		Lysine	Methionine + Cysteine	Threonine
8	1.040	965.91	864.45	881
12	0.963	965.91	860.48	881
16	0.930	979.18	856.52	881
20	0.873	727.00	741.39	881
Mean		909.5	830.71	881

).

4. Discussion

4.1. Egg Cost and Zootechnical Results

The iLys requirement of 943.7 mg/hen-day found in diet 13, corresponding to the treatment with lower FCK (Table 4), is similar to that reported by Schneider [26], who indicated 942 mg/hen-day for maximum production, and is above the values observed by Rostagno et al. [27], Schmidt et al. [28], and Kakhki et al. [29] of 807 mg/hen-day, 824 mg/hen-day, and 848 mg/hen-day, respectively.

The iMetCys requirement of 858.3 mg/hen-day found in diet 13 is above the values of 747.46 mg/hen-day, 786.51 mg/hen-day, 791 mg/hen-day, and 811 mg/hen-day reported for maximum production by Macelline et al. [9], Polese et al. [30], Schmidt et al. [28], and Rostagno et al. [27], respectively.

The iThr requirement of 876.8 mg/hen-day observed in diet 13 is above the values reported for maximum production by Agustini et al. [31], Schmidt et al. [32], and Rostagno et al. [27] of 516.26 mg/hen-day, 525.04 mg/hen-day, and 621 mg/hen-day, respectively.

For the best FCK treatment (diet 13), the CR was 2.16 ± 0.14 kg feed/kg egg (Table 4) for H&N Brown SCLHs at 20 weeks of production (111 weeks of age). This result is lower than that obtained by Sariozkan et al. [33], who found 2.2 to 2.4 kg feed/kg egg at week 20 of production in Hy-Line Brown SCLHs between 81 and 92 weeks of age. The feed conversion ratio result for diet 13 found at 16 weeks (2.18 ± 0.17 kg feed/kg egg) is higher than the 1.97 kg feed/kg egg reported for Isa Brown hens at 16 weeks (103–106 weeks of age) [34] and 2.04 kg feed/kg of egg reported for H&N Brown SCLHs at 14 weeks of production (81–95 weeks of age) [35].

A good level of amino acids in the diet, especially sulfur amino acids (Met + Cys), is essential in the second production cycle since they directly influence egg size [36]. The difference in egg sizes could also contribute to making diet 13 the most cost-effective treatment. The percentage of eggs weighing 60 g or more (94%) was higher than that of diet 2 (92%) [12], which presented the highest FCK.

Finally, the results of diet 13 regarding egg production and CR at week 20 [12] show the importance of each EAA in meeting the nutritional needs of H&N Brown SCLHs. From 8 weeks of egg production, the productive levels increased with the consumption of Lys in the diet because this EAA is considered physiologically crucial for the maintenance, growth, and production of laying hens, and its primary function is the synthesis of muscle protein. In addition, egg production responds to high levels of Lys, possibly due to the increase in the concentration of plasma albumin, which is the main protein required by the body for the synthesis of egg protein in the oviduct [36].

4.2. ANN Model to Analyze Egg Cost as a Function of EAAs and Time

The ANN models accurately described the nonlinear behavior for H&N Brown SCLHs compared to the other models. These models also enabled effective prediction of FCK based on iLys, iMetCys, iThr, and time (Table 5). The fitting results were better for both the training and validation data sets. The AIC values also showed that the ANN models presented a better generalization than the polynomial models.

The ANN models were appropriate for estimating FCK, as confirmed by the random and balanced distribution of the residuals around zero, with 98.2% of them ranging from −0.12 to 0.12 USD/kg egg (Figure 2), despite the non-normality of the residuals (p-value < 0.05) (Table 5).

According to previous results [12], the ANN models were more accurate in describing FKC than egg production based on EAAs and time in H&N Brown SCLHs. Although this is the first study of its kind, applying ANNs to simultaneously analyze the influence of nutritional factors and time on feed costs, other studies have reported that ANN models are more precise than other nonlinear regression models in estimating zootechnical parameters [37,38].

The versatility of ANNs allows for estimating the weekly, overall, or final nutritional requirements in terms of costs for the second production cycle. The best FCK estimations covered intermediate levels of EAAs among those evaluated in the experimental design (Table 1). Thus, the ANN model estimated levels of iLys for low FCK values at 20 weeks (Figure 3) comparable to the 942 mg/hen-day recommended by Schneider [26] for Shaver Brown SCLHs. On the contrary, other authors recommend lower levels: 713 mg/hen-day for hens in the post-molting phase [25], or 751 mg/hen-day [9], 848 mg/hen-day [29], and 807 mg/hen-day for laying hens in the first production cycle [27].

Combinations of iLys, iMetCys, and iThr to minimize FCK at different weeks can be found in Figure 3. For week 8, FCK values less than 1.076 USD/kg egg can be achieved with 943–1000 mg/hen-day of iLys, 800–1000 mg/hen-day of iMetCys, and 717–881 mg/hen-day of iThr. For week 12, the recommended levels are 943–1000 mg/hen-day of iLys, 853–961 mg/hen-day of iMetCys, and 881 mg/hen-day of iThr, which can result in FCK values less than 0.973 USD/kg egg. For week 16, the adequate levels are 943–1000 mg/hen-day of iLys, 853–961 mg/hen-day of iMetCys, and 881 mg/hen-day of iThr, which can lead to FCK values less than 0.948 USD/kg egg. For week 20, several possible combinations of EAA intakes can produce FCK values below 0.922 USD/kg egg, as shown in Figure 3. For example, some options could be 728 mg/hen-day of iLys, 661–991 mg/hen-day of iMetCys, and 552 mg/hen-day of iThr, or 727 mg/hen-day of iLys, 700–900 mg/hen-day of iMetCys, and 881 mg/hen-day of iThr.

The limiting amino acid concept is based on the interruption of protein synthesis due to amino acid deficiency in the provided diet. For birds, methionine is the first limiting amino acid, not only due to the high demand for cysteine to form feathers and methionine to form ovalbumin but also due to the composition of the diet generally used for this species [39]. As glycogenic agents, methionine and cysteine are metabolized to pyruvic acid via succinyl-CoA and participate in gluconeogenesis—the process of making glycogen in the liver and muscle—improving the body recovery of the birds [36].

The estimated iThr at 20 weeks is also different from the 562 mg/hen-day for Shaver Brown hens (75–90 weeks of age) [31], 462 mg/hen-day for Lohmann Brown SCLHs (79–95 weeks of age) [32], and 499 mg/hen-day for Hy-Line commercial layers [25]. These reported levels are lower than the 876.8 mg/hen-day of iThr recommended in this study for H&N Brown SCLHs.

Threonine becomes more relevant the older the hens are because its maintenance ratio increases [40]. Therefore, adequate Thr supplementation allows the bird to express its maximum production potential. Hence, its nutritional demand is important to consider in egg production, albeit indirectly [41]. The large membrane carriers are found in the ileum, suggesting that this is a crucial site of Thr uptake. Likewise, the intestinal mucosa develops by growing taller and denser villi, which means more epithelial cells and improved digestion and absorption of nutrients [42].

The ANN model shows how each EAA affects different FCK outcomes (Figure 3), indicating that it is necessary to validate scenarios of technical and economic interest to the producers and studies where other nutritional requirements can be used for different production stages.

The optimal levels of Lys, Met + Cys, and Thr for SCLHs have been inconsistent across different studies despite being the most critical EAAs for laying hens. This study demonstrated the importance of the interactions among the EAAs for predicting the egg cost in laying hens, using ANN models as a novel approach.

4.3. Optimization

The minimum FCK decreased between weeks 8 and 20 from 1.040 to 0.873 USD/kg egg. Digestible iLys and iMetCys levels diminished between 8 and 20 weeks from 965.91 mg/hen-day to 727 mg/hen-day and from 864.45 to 741.39 mg/hen-day, respectively. Digestible iThr levels were 881 mg/hen-day for all weeks analyzed (Table 7). At 20 weeks of production, the iMetCys/iLys ratio was 102%, and the iThr/iLys ratio was 121%.

H&N Brown SCLHs showed Thr requirements of 881 mg/hen-day, higher than those of laying hens in the first production cycle, to achieve excellent egg production and feed conversion results with the lowest FCK. Thr supplementation enables animals to make use of proteins and minor amino acids in feed [31]. Protein synthesis and body protein turnover depend on Thr. In addition, Thr and serine are essential for feather formation, accounting for 20% of EAA requirements [40]. Mucin production, necessary for intestinal health, nutrient absorption, and antibody production, also requires Thr [43].

The iThr/iLys ratio at 20 weeks of 121.18% for H&N Brown SCLHs is higher than the EAA estimation of 77% for laying hens during the first cycle of production [27]. The iMetCys/iLys ratio at 20 weeks of 101.98% for H&N Brown SCLHs is also higher than the 98% reported by Rostagno et al. [27].

The mean optimal requirements (Table 7) are close to the EAA intakes in diet 13. On diet 13, the observed FCK between weeks 8 and 20 decreased by 7.92% (Table 3), corresponding to an increment of 2.45% in egg production [12] and a decrease of 3.36% in CR (Table 4). When comparing the observed and estimated FCK, we found percentage errors of less than 2% between weeks 12 and 20, reflecting the accuracy of the ANN model.

5. Conclusions

According to this study, the feed cost per kilogram of egg for H&N Brown second-cycle laying hens is better described by artificial neural networks than by multivariate polynomial models. Artificial neural networks can also estimate essential amino acid intakes to minimize feed costs, offering excellent egg production and conversion ratio results.

The requirements of lysine, methionine + cysteine, and threonine in H&N Brown second-cycle laying hens to minimize the feed cost per kilogram of egg differ from those defined to maximize egg production when models based on artificial neural networks are used. This modeling methodology will allow producers to design profitable and productive diets according to the changing scenarios of the poultry sector.