Nonlinear Feed Formulation For Broiler

Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 Nonlinear Feed Formulation For Broiler: Modeling And Optimization Marcel Alessandro de Almeida1; Manoel Garcia Neto2; Max José de Araujo Faria Junior2; Marcos Franke Pinto2; Leda Gobbo de Freitas Bueno2 ¹Master by the Graduate Program in Animal Science - UNESP, Araçatuba. ² Professor of the Department of Support, Production and Animal Health - UNESP / Araçatuba, SP Brazil - Rua. Clovis Pestana, 793 - Bairro Dona Amélia - Araçatuba / SP Zip code 16050-680 phone: 55 (18) 3636-1400 Corresponding author: [email protected] Summary The current scenario requires the application of new computational tools for the feed formulation strategy that uses mathematical modeling in decision making. Noteworthy is the nonlinear programming, which aims not only to formulate a diet that meets the needs of the animal, but also the minimum cost and the maximum profit margin. Thus, the work aimed to validate the use of the nonlinear model (NLM), with maximization of the economic return, through estimates of animal performance and feed costs, according to the price variation of the kg of the broiler (price historical average of 2009 and 2010), the phases of creation and sex. For this purpose, 480 broiler broiler chickens, 240 males and 240 females of the same strain (Cobb 500) were used, from 1 to 56 days of age. The experimental design was entirely randomized, totaling 6 treatments (increasing or decreasing the average historical price of live chicken by 25% or 50%), with 4 replicates and 10 broiler chickens per experimental plot. Performance (weight gain and feed consumption), total energy consumption and profit margin were evaluated. Regarding the formulation principle (Linear and Nonlinear), the performance was very similar in relation to the studied parameters. However, when simulated values of 50% below the historical average, performance was significantly impaired in this specific condition. However, due to the profit margin, it demonstrated that the principle of nonlinear formulation allows to significantly reduce losses (P <0.05), mainly in unfavorable conditions of the price of chicken in the market. It is concluded that the nonlinear principle is more appropriate, since the requirements of all nutrients are automatically adjusted by the mathematical model and with the premise of increasing profitability, different from the linear one, which is to achieve maximum performance and not is directly related to the economic factor. Keywords: data modeling, nonlinear programming, nutritional strategies, optimization, profitability. 1. INTRODUCTION The industry's search for a constant increase in productivity and profit, which involves not only greater slaughter weight at a younger age, but also higher carcass and cut yields; in addition to the International Educative Research Foundation and Publisher © 2020 pg. 262 International Journal for Innovation Education and Research www.ijier.net Vol:-08 No-12, 2020 growing consumer demand for lean meat intake, it imposes a challenge on feed formulators. This is because dealing with cost-benefit relations presupposes the integration of biological and economic aspects [3]. The commercial formulation of diets for broilers consists of combining ingredients in appropriate proportions to achieve the appropriate and desired nutritional profile, aiming at the optimum level between performance and cost and, consequently, maximum profitability [10]. An alternative to help in making decisions and defining better and more economical products is the use of computational modeling. This methodology seeks to transform pertinent concepts and knowledge into mathematical equations and implements them through logical processes, simulating real situations on a computer [14]. Efficiency in feed formulation is one of the needs of the animal production industry. Animal performance and development are directly linked to food intake and in order to meet the animal's requirement at a certain stage of production, it is very important that the diet is formulated efficiently [17] [19]. To improve the commercial production process, precision models of feed consumption, growth and carcass yields are of crucial importance for the economy [20]. Thus, the linear model (LM), by defining only the minimum cost of the feed, will not necessarily allow a maximum profit, hence its great limitation. This limitation promoted the development of the nonlinear concept, which seeks the best gain rates, however, allying the minimum cost diets that meet nutritional requirements [8]. The present study aimed to validate the use of a nonlinear simulation spreadsheet, with maximization of the economic return, through estimates of poultry performance and production costs, according to the variation in the price of kg of broiler and the phases from creation. 2. MATERIAL AND METHODS The experiments were carried out in the Animal Science Sector of the Faculty of Veterinary Medicine of Araçatuba (FMVA), at Universidade Estadual Paulista (UNESP). Two experiments I (females) and II (males) were carried out, consisting of diets formulated according to the linear (minimum cost) and nonlinear (maximum profit) systems. Commercial broiler chickens (Cobb 500) were used, with 240 males and 240 females, from 1 to 56 days. The experiment was approved by the Commitee for Ethical Use Animals (CEUA) of São Paulo State University (UNESP) at campus Faculty of Veterinary Medicine (FMVA) at campus Araçatuba / SP under protocol number 008872012. The experimental design was completely randomized, totaling 6 treatments for each experiment, and four repetitions according to the price per kg of chicken paid (normal LM, + 50%, + 25%, -50%, -25% and normal NLM) . Subsequently, to assess the economic viability, a completely randomized design was used, with 10 International Educative Research Foundation and Publisher © 2020 pg. 263 Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 treatments and four replications. To house the broiler chickens, a masonry shed (7.85 x 45.70 m) was used, with East-West orientation, air-conditioned by an adiabatic evaporative cooling system with negative pressure ventilation, covered with tiles made of insulating material (expanded polystyrene) disposed between reflective metal plates. Inside, the chickens were placed in boxes, with a tubular feeder and pendulum drinker for each, with dimensions of 1.4 x 3.0 m, which were constituted in the experimental plots, with a bed of wood shavings and an animal density 2.38 chickens/m². One-day-old broiler chickens were weighed and randomly distributed in 48 boxes (four replicates with 10 chickens per treatment). As initial heating sources, porcelain cones with electrical resistance of 400W were used, with one remaining in each compartment during the first 15 days of creation. The diets were formulated based on corn, soybean meal, soybean oil, vitamin supplement, mineral supplement, limestone and dicalcium phosphate, using the recommendations of [16], according to the linear (minimum cost ration) and nonlinear ( maximum profit ration) according to the mathematical model of [5] that determined the feeding strategy for males and females of broilers, defined by the Practical Program for Feed Formulation (PPFR) (Tables 1 and 2). The results were subjected to analysis of variance to verify the effects of treatments according to the PROC GLM system procedures [18]. In order to verify the significance of the differences between treatment means, the T test (LSD) was applied. As there are differences between the growth rates for males and females, with different nutritional recommendations, and due to the different formulations imposed by nonlinear programming, the possibility of using a factorial scheme was disregarded [15]. According to [4], the responses for the production of broilers, corresponding to age and the energy content of the diet, understood as being "nutritional density", are defined through the quadratic function, as to the equations. The complete models adjusted for broilers from 1 to 20 days1: 𝑭𝒆𝒎𝒂𝒍𝒆 𝒍𝒊𝒗𝒆 𝒘𝒆𝒊𝒈𝒉𝒕 = −𝟐𝟔𝟐𝟗, 𝟑𝟗𝟐𝟔𝟏𝟔 + 𝟏, 𝟕𝟖𝟔𝟏𝟕𝟑 ∗ 𝑴𝑬 − 𝟏𝟓, 𝟑𝟐𝟓𝟑𝟗𝟒 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟎𝟐𝟗𝟖 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟎𝟗𝟓𝟒𝟕 ∗ 𝑨 ∗ 𝑴𝑬 − 𝟏, 𝟎𝟑𝟑𝟏𝟒 ∗ 𝑨² 𝑴𝒂𝒍𝒆 𝒍𝒊𝒗𝒆 𝒘𝒆𝒊𝒈𝒉𝒕 = −𝟑𝟑𝟓𝟒, 𝟑𝟑𝟎𝟗𝟏𝟔 + 𝟐, 𝟐𝟕𝟓𝟏𝟖𝟑 ∗ 𝑴𝑬 − 𝟐𝟔, 𝟎𝟐𝟒𝟗𝟔𝟒 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟎𝟑𝟖 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟏𝟐𝟕𝟔𝟖 ∗ 𝑨 ∗ 𝑴𝑬 − 𝟏, 𝟐𝟑𝟖𝟕𝟒𝟏 ∗ 𝑨² 𝑭𝒆𝒎𝒂𝒍𝒆 𝒇𝒆𝒆𝒅 𝒄𝒐𝒏𝒔𝒖𝒎𝒑𝒕𝒊𝒐𝒏 = −𝟐𝟏𝟒𝟏, 𝟏𝟎𝟗𝟗𝟖𝟐 + 𝟏, 𝟑𝟗𝟔𝟐𝟒𝟗 ∗ 𝑴𝑬 + 𝟐𝟔, 𝟒𝟑𝟒𝟗𝟒𝟏 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟎𝟐𝟐𝟑 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟎𝟕𝟓𝟓𝟔 ∗ 𝑨 ∗ 𝑴𝑬 + 𝟐, 𝟑𝟕𝟔𝟗𝟎𝟓 ∗ 𝑨² 𝑴𝒂𝒍𝒆 𝒇𝒆𝒆𝒅 𝒄𝒐𝒏𝒔𝒖𝒎𝒑𝒕𝒊𝒐𝒏 = −𝟐𝟕𝟑𝟑, 𝟑𝟎𝟔𝟑𝟓𝟖 + 𝟏, 𝟕𝟖𝟐𝟓𝟕𝟔 ∗ 𝑴𝑬 + 𝟐𝟔, 𝟒𝟏𝟎𝟔𝟓𝟐 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟎𝟐𝟖𝟓 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟎𝟖𝟖𝟖𝟔 ∗ 𝑨 ∗ 𝑴𝑬 + 𝟐, 𝟖𝟏𝟗𝟏𝟕𝟏 ∗ 𝑨² 1 ME and A represent the Metabolizable Energy and the Age, respectively. The complete models adjusted for broilers from 21 to 56 days1: 𝑭𝒆𝒎𝒂𝒍𝒆 𝒍𝒊𝒗𝒆 𝒘𝒆𝒊𝒈𝒉𝒕 = −𝟑𝟏𝟗𝟑𝟓 + 𝟐𝟎, 𝟎𝟏𝟔𝟒𝟓𝟑 ∗ 𝑴𝑬 + 𝟖𝟑, 𝟒𝟒𝟓𝟐𝟎𝟏 ∗ 𝑨 − 𝟎, 𝟎𝟑𝟐𝟑𝟕 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟎𝟑𝟕𝟔𝟕 ∗ 𝑨 ∗ 𝑴𝑬 − 𝟎, 𝟐𝟑𝟐𝟓𝟒𝟖 ∗ 𝑨² 𝑴𝒂𝒍𝒆 𝒍𝒊𝒗𝒆 𝒘𝒆𝒊𝒈𝒉𝒕 = −𝟐𝟓𝟕𝟖𝟏 + 𝟏𝟓, 𝟗𝟖𝟖𝟔𝟎𝟗 ∗ 𝑴𝑬 + 𝟔𝟒, 𝟕𝟎𝟔𝟑𝟖 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟐𝟔𝟎𝟖 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟏𝟓𝟎𝟎𝟔 ∗ 𝑨 ∗ 𝑴𝑬 − 𝟎, 𝟐𝟏𝟑𝟖𝟏𝟕 ∗ 𝑨² 𝑭𝒆𝒎𝒂𝒍𝒆 𝒇𝒆𝒆𝒅 𝒄𝒐𝒏𝒔𝒖𝒎𝒑𝒕𝒊𝒐𝒏 = −𝟒𝟗𝟗𝟗𝟖 + 𝟑𝟏, 𝟏𝟗𝟔𝟗𝟏𝟑 ∗ 𝑴𝑬 + 𝟐𝟏𝟗, 𝟑𝟓𝟎𝟐𝟓𝟕 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟒𝟗𝟗𝟗 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟑𝟒𝟕𝟖𝟑 ∗ 𝑨 ∗ 𝑴𝑬 − 𝟎, 𝟕𝟒𝟗𝟕𝟔𝟑 ∗ 𝑨² 𝑴𝒂𝒍𝒆 𝒇𝒆𝒆𝒅 𝒄𝒐𝒏𝒔𝒖𝒎𝒑𝒕𝒊𝒐𝒏 = −𝟑𝟕𝟓𝟒𝟕 + 𝟐𝟒, 𝟎𝟓𝟔𝟎𝟔𝟒 ∗ 𝑴𝑬 + 𝟐𝟓𝟕, 𝟓𝟎𝟔𝟎𝟒𝟗 ∗ 𝑨 − 𝟎, 𝟎𝟎𝟑𝟖𝟏 ∗ 𝑴𝑬𝟐 + 𝟎, 𝟎𝟒𝟐𝟐𝟒𝟏 ∗ 𝑨 ∗ 𝑴𝑬 − 𝟎, 𝟕𝟗𝟐𝟗𝟗𝟔 ∗ 𝑨² 1 ME and A represent the Metabolizable Energy and the Age, respectively. International Educative Research Foundation and Publisher © 2020 pg. 264 International Journal for Innovation Education and Research www.ijier.net Vol:-08 No-12, 2020 The objective functions for profit margin (PM) for males (PMm) and females (PMf) were obtained1: PMm = -0.879527 + 0.090166 × A-0.019683 × PM-0.000576 × A ^ 2 + 0.001738 × PM × A PMf = -0.613252 + 0.075129 × A-0.012823 × PM-0.000615 × A ^ 2 + 0.00135 × PM × A 1 A represent the Age. The broilers were evaluated through their body weight gain, feed intake and feed conversion index. Weight gain (g / broiler / period), feed intake (g / broiler / period) and feed conversion were verified at 21°, 42° and 56° days of age. From these data, the bioeconomic index (IBE), adapted from [6], Economic efficiency (EFE) adapted by [7] and Bioeconomic Energy Conversion (BEC), was calculated in order to reduce the distortions made by the indices. As they do not consider energy in the evaluation of economic efficiency, IBE and EFE would not be appropriate, due to the fact that in the nonlinear model diets with different energy levels are formulated in the same creation phase, which does not occur in the linear model, which formulates diets with defined energy requirements, that is why in this work the BEC (Bioeconomic Energy Conversion) index was proposed in order to evaluate this new formulation principle. The BEC Eq formula (1) integrates the total energy intake (TEI) in Megacalories (Mcal), the weighted cost of the feed (WCF) in (R$/kg), the weight gain (WG) in (kg) and the price of live chicken (PC)(R$/kg). 𝑩𝑬𝑪 = 𝑇𝐸𝐼×𝑊𝐶𝐹 𝑊𝐺×𝑃𝐶 Eq (1) (𝑀𝑐𝑎𝑙/𝑘𝑔) It is observed that the cost per kg of the feed should be the weighted (WCF) Eq (2), because this way an average value of the feed cost is obtained with greater accuracy. Therefore the weighted cost for the experiment was: 𝑾𝑪𝑭 = IFC×21+GFC×21+TFC×14 Eq(2) 56 Where: IFC = initial feed cost; GFC = growth feed cost; TFC = termination feed cost. In relation to the other indexes, EFE [7], it was calculated in relation to the income obtained by weight gain and the cost invested in food in each period Eq (3), thus allowing an economic view of productivity in our market [7] through the currency of the Federal Republic of Brazil (R$) and the IBE [6] [12], used it to perform the calculation the average weight gain in the period, the relationship between the price of 1kg of feed (PF) and the sale price of 1kg of live chicken (PC) and the average feed consumption (FC), in each treatment Eq (4) . 𝑬𝑭𝑬 = 𝑊𝑒𝑖𝑔ℎ𝑡 𝑔𝑎𝑖𝑛 𝑖𝑛𝑐𝑜𝑚𝑒 𝑓𝑒𝑒𝑑 𝑐𝑜𝑠𝑡 𝑃𝐹 (𝑅$/𝑅$) 𝑰𝑩𝑬 = weight gain − ⌊( ) × FC⌋ (kg) 𝑃𝐶 International Educative Research Foundation and Publisher © 2020 Eq (3) Eq(4) pg. 265 Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 Table 1 - Composition of the feed ingredients (%) and the calculated nutrient content of the diet (%), according to the stages and requirements for females. 0.82 % 0.582 0.000 61.881 0.000 34.293 1.581 0.447 0.109 0.189 0.000 0.824 0.676 0.000 Nonlinear spreadsheet Price per kilogram of Broiler 2.05 1.64 1.23 % % % 0.671 0.657 0.624 0.000 0.000 0.000 57.258 58.692 62.135 3.803 3.096 1.398 34.618 33.930 32.277 1.744 1.722 1.670 0.476 0.471 0.458 0.209 0.214 0.225 0.254 0.249 0.237 0.056 0.056 0.055 0.862 0.858 0.847 0.721 0.714 0.698 0.000 0.000 0.000 2.877 20.865 0.805 0.404 0.801 0.197 0.336 1.362 1.086 0.484 0.771 0.229 0.705 1.320 0.878 0.819 1.704 0.529 0.959 1.618 137.869 2.970 20.152 0.831 0.417 0.765 0.201 0.364 2.107 1.121 0.519 0.796 0.218 0.728 1.257 0.840 0.781 1.642 0.507 0.917 1.547 147.391 Ingredients Feed cost Inert Corn Soy oil Soybean mean -45% Dicalcium phosphate Common salt L-Lysine HCl DL-Methionine L Threonine Calcitic limestone Polimax F-pre initial (Fatec) Polimax F-3 finishing (Fatec) Calculated composition Metabolizable Energy (Kcal kg-1) Crude protein (%) Calcium (%) Available phosphorus (%) Potassium (%) Sodium(%) Chlorine (%) Linoleic acid Dig. Lysine Dig. Methionine Dig. Methionine + Cystine Dig. Tryptophan Dig. Threonine Dig. Arginine Dig. Valine Dig. Isoleucine Dig. Leucine Dig. Histidine Dig. Phenylalanine Dig. Phenylalanine+Tyrosine Energy:Protein Ratio 3.040 20.613 0.850 0.427 0.785 0.205 0.368 2.971 1.147 0.535 0.814 0.225 0.746 1.297 0.860 0.804 1.662 0.517 0.940 1.584 147.495 3.070 20.805 0.858 0.431 0.794 0.207 0.370 3.331 1.158 0.541 0.822 0.228 0.753 1.314 0.869 0.814 1.670 0.522 0.949 1.600 147.538 Finisher (43 a 56 days of age) Grower (22 a 42 days of age) Starter (1 a 21 days of age) Linear spreadsheet 2.46 % 0.678 0.000 56.507 4.173 34.978 1.755 0.479 0.206 0.257 0.056 0.864 0.725 0.000 % 0.662 0.000 56.893 3.566 35.313 1.724 0.472 0.176 0.242 0.041 0.856 0.716 0.000 0.82 % 0.511 1.890 67.259 0.000 27.909 1.240 0.382 0.000 0.084 0.000 0.721 0.515 0.000 Nonlinear spreadsheet Price per kilogram of Broiler 2.46 2.05 1.64 1.23 % % % % 0.535 0.535 0.531 0.531 0.000 0.000 0.000 0.000 73.534 73.556 72.664 72.664 0.012 0.000 0.000 0.000 23.085 23.074 24.047 24.047 1.330 1.330 1.318 1.318 0.398 0.398 0.397 0.397 0.192 0.193 0.158 0.158 0.142 0.142 0.132 0.132 0.015 0.015 0.000 0.000 0.756 0.756 0.753 0.753 0.535 0.535 0.533 0.533 0.000 0.000 0.000 0.000 3.085 20.906 0.863 0.433 0.798 0.208 0.371 3.519 1.164 0.545 0.826 0.230 0.757 1.323 0.873 0.819 1.674 0.524 0.954 1.608 147.559 3.050 21.041 0.853 0.428 0.806 0.206 0.362 3.201 1.151 0.533 0.817 0.232 0.748 1.335 0.881 0.826 1.689 0.529 0.963 1.623 144.958 2.907 18.252 0.668 0.333 0.699 0.171 0.275 1.418 0.853 0.352 0.614 0.197 0.620 1.135 0.774 0.711 1.551 0.471 0.843 1.423 159.250 3.010 17.125 0.691 0.345 0.644 0.177 0.315 1.491 0.883 0.385 0.636 0.178 0.574 1.031 0.718 0.651 1.479 0.441 0.781 1.319 175.753 3.010 17.125 0.691 0.345 0.644 0.177 0.315 1.491 0.883 0.385 0.636 0.178 0.574 1.031 0.718 0.651 1.479 0.441 0.781 1.319 175.753 3.020 16.808 0.694 0.346 0.628 0.177 0.323 1.501 0.886 0.391 0.638 0.173 0.576 1.003 0.703 0.634 1.456 0.433 0.764 1.289 179.674 3.021 16.811 0.694 0.346 0.628 0.177 0.323 1.507 0.886 0.391 0.638 0.173 0.576 1.003 0.703 0.634 1.457 0.433 0.764 1.290 179.674 Linear spreadsheet % 0.614 0.000 65.413 4.225 26.814 1.440 0.427 0.158 0.166 0.013 0.777 0.567 0.000 0.82 % 0.464 6.227 66.041 0.000 25.036 1.101 0.351 0.000 0.065 0.000 0.662 0.000 0.517 Nonlinear spreadsheet Price per kilogram of Broiler 2.46 2.05 1.64 1.23 % % % % 0.506 0.506 0.501 0.495 0.000 0.000 0.000 0.000 74.816 74.816 72.777 70.427 0.000 0.000 0.000 0.000 22.003 22.003 24.184 26.699 1.230 1.230 1.204 1.174 0.382 0.382 0.378 0.375 0.167 0.167 0.090 0.000 0.118 0.118 0.095 0.069 0.000 0.000 0.000 0.000 0.725 0.725 0.716 0.706 0.000 0.000 0.000 0.000 0.561 0.561 0.556 0.551 3.200 17.810 0.735 0.367 0.674 0.188 0.331 3.655 0.939 0.423 0.676 0.189 0.610 1.094 0.746 0.685 1.498 0.455 0.813 1.372 179.674 2.800 16.839 0.604 0.302 0.643 0.158 0.255 1.376 0.777 0.314 0.559 0.180 0.571 1.039 0.714 0.652 1.447 0.436 0.778 1.312 166.285 2.986 17.957 0.644 0.322 0.686 0.168 0.272 1.468 0.829 0.335 0.596 0.192 0.609 1.108 0.762 0.696 1.543 0.465 0.829 1.399 166.285 3.013 17.111 0.650 0.324 0.646 0.170 0.291 1.494 0.836 0.350 0.602 0.179 0.576 1.036 0.721 0.654 1.485 0.443 0.784 1.324 176.079 3.036 16.377 0.655 0.327 0.612 0.171 0.308 1.517 0.843 0.362 0.606 0.168 0.547 0.973 0.687 0.617 1.434 0.424 0.745 1.259 185.394 3.036 16.377 0.655 0.327 0.612 0.171 0.308 1.517 0.843 0.362 0.606 0.168 0.547 0.973 0.687 0.617 1.434 0.424 0.745 1.259 185.394 Linear spreadsheet % 0.000 66.737 4.733 25.076 1.359 0.415 0.000 0.165 0.153 0.012 0.749 0.000 0.600 3.250 17.130 0.701 0.350 0.646 0.183 0.325 3.942 0.902 0.403 0.649 0.180 0.586 1.043 0.718 0.655 1.455 0.439 0.781 1.318 189.726 Vitamin-mineral supplements used in diets in three rearing stages (quantity / kg of product) included: pre Initial: vit. A - 1,835,000 I.U. vit. D3 - 335,000 I.U. vit. E - 2,835 mg; vit. K3 - 417 mg; vit. B1 - 335 mg; vit. B2 - 1,000 mg; vit. B6 - 335 mg; vit. B12 - 2,500 mcg; folic acid - 135 mg; biotin - 17 mg; niacin - 6,670 mg; calcium pantothenate - 1,870 mg; Cu - 1,000 mg; Co - 35 mg; I - 170 mg; Fe - 8,335 mg; Mn - 10,835mg; Zn - 8,335 mg; Se - 35 mg; Choline Chloride 50% - 135,000 mg; Methionine - 267,000 mg; Coccidiostatic - 13,335 mg; Growth Promoter - 16,670 mg; Antioxidant - 2,000 mg. Termination: vit. A - 1,670,000 I.U. vit. D3 - 335,000 I.U. vit. E - 2,335 mg; vit. K3 - 400 mg; vit. B1 - 100 mg; International Educative Research Foundation and Publisher © 2020 pg. 266 International Journal for Innovation Education and Research www.ijier.net Vol:-08 No-12, 2020 vit. B2 - 800 mg; vit. B6 - 200 mg; vit. B12 - 2,000 mcg; folic acid - 67 mg; biotin - 7 mg; niacin - 5,670 mg; calcium pantothenate - 2,000 mg; Cu - 2,000 mg; Co - 27 mg; I - 270 mg; Fe - 16,670 mg; Mn - 17,335 mg; Zn - 12,000 mg; Se - 70 mg; Choline Chloride 50% - 100,000mg; Methionine - 235,000mg; Antioxidant - 2,000 mg. Table 2 - Composition of feed ingredients (%) and calculated nutrient content of the diet (%), according to the stages and requirements for males. Starter (1 a 21 days of age) Ingredients 0.82 % Feed cost Inert Corn Soy oil Soybean mean -45% Dicalcium phosphate Common salt L-Lysine HCl DL-Methionine L Threonine Calcitic limestone Polimax F-pre initial (Fatec) Polimax F-3 finishing (Fatec) Calculated composition Metabolizable Energy (Kcal kg-1) Crude protein (%) Calcium (%) Available phosphorus (%) Potassium (%) Sodium(%) Chlorine (%) Linoleic acid Dig. Lysine Dig. Methionine Dig. Methionine + Cystine Dig. Tryptophan Dig. Threonine Dig. Arginine Dig. Valine Dig. Isoleucine Dig. Leucine Dig. Histidine Dig. Phenylalanine Dig. Phenylalanine+Tyrosine Energy:Protein Ratio Nonlinear spreadsheet Price per kilogram of Broiler 1.23 1.64 2.05 2.46 % % % % Growero (22 a 42 days of age) Linear spreadsheet % 0.82 % Nonlinear spreadsheet Price per kilogram of Broiler 1.23 1.64 2.05 2.46 % % % % Finisher (43 a 56 days of age) Linear spreadsheet % 0.82 % Nonlinear spreadsheet Price per kilogram of Broiler 1.23 1.64 2.05 2.46 % % % % Linear spreadsheet % 0.599 0.659 0.000 0.000 63.092 57.260 0.000 3.004 32.679 35.304 1.707 1.808 0.472 0.496 0.215 0.208 0.233 0.259 0.049 0.056 0.874 0.895 0.678 0.708 0.000 0.000 0.686 0.000 54.458 4.371 36.662 1.852 0.507 0.199 0.269 0.057 0.904 0.721 0.000 0.696 0.000 53.392 4.891 37.179 1.868 0.511 0.195 0.272 0.057 0.908 0.726 0.000 0.702 0.000 52.852 5.155 37.441 1.877 0.513 0.194 0.274 0.057 0.909 0.729 0.000 0.677 0.000 54.196 4.091 37.301 1.830 0.503 0.167 0.256 0.042 0.898 0.716 0.000 0.511 0.312 62.110 0.000 30.846 1.257 0.398 0.000 0.116 0.000 0.723 0.496 0.000 0.542 0.000 68.270 0.000 28.368 1.358 0.412 0.137 0.161 0.000 0.769 0.524 0.000 0.544 0.000 68.867 0.000 27.716 1.366 0.414 0.161 0.168 0.010 0.772 0.526 0.000 0.556 0.000 69.663 0.293 26.489 1.393 0.418 0.213 0.186 0.034 0.779 0.531 0.000 0.586 0.000 66.603 1.859 27.910 1.436 0.429 0.202 0.196 0.034 0.788 0.542 0.000 0.648 0.000 60.152 5.161 30.905 1.526 0.453 0.180 0.218 0.034 0.805 0.567 0.000 0.507 0.000 66.476 0.000 30.535 1.215 0.400 0.000 0.105 0.000 0.725 0.000 0.543 0.517 0.000 70.242 0.000 26.512 1.264 0.395 0.145 0.148 0.000 0.742 0.000 0.551 0.517 0.000 70.242 0.000 26.512 1.264 0.395 0.145 0.148 0.000 0.742 0.000 0.551 0.525 0.000 72.193 0.000 24.387 1.291 0.399 0.222 0.171 0.033 0.750 0.000 0.555 0.534 0.000 71.275 0.476 24.806 1.303 0.402 0.219 0.174 0.033 0.753 0.000 0.559 0.631 0.000 61.183 5.717 29.423 1.438 0.438 0.183 0.207 0.034 0.778 0.000 0.600 2.888 20.397 0.851 0.425 0.775 0.206 0.371 1.374 1.126 0.519 0.799 0.221 0.732 1.273 0.852 0.791 1.664 0.514 0.930 1.567 141.590 3.071 21.500 0.905 0.452 0.823 0.220 0.386 3.599 1.197 0.563 0.850 0.238 0.778 1.369 0.898 0.846 1.709 0.538 0.982 1.656 142.843 3.092 21.645 0.912 0.455 0.830 0.221 0.388 3.864 1.206 0.568 0.856 0.240 0.784 1.382 0.904 0.853 1.715 0.541 0.989 1.667 142.872 3.103 21.719 0.915 0.457 0.833 0.222 0.389 3.998 1.210 0.570 0.859 0.241 0.787 1.388 0.908 0.856 1.718 0.542 0.993 1.673 142.886 3.050 21.719 0.899 0.449 0.834 0.218 0.378 3.448 1.189 0.554 0.844 0.241 0.773 1.389 0.910 0.857 1.727 0.544 0.995 1.677 140.432 2.800 19.176 0.678 0.338 0.738 0.177 0.284 1.343 0.917 0.392 0.661 0.210 0.652 1.211 0.813 0.754 1.596 0.491 0.887 1.496 146.015 2.959 18.718 0.717 0.357 0.710 0.183 0.321 1.439 0.969 0.432 0.698 0.200 0.630 1.154 0.786 0.722 1.576 0.479 0.857 1.446 158.082 2.966 18.506 0.718 0.358 0.700 0.184 0.326 1.446 0.971 0.436 0.700 0.197 0.631 1.135 0.776 0.711 1.560 0.473 0.845 1.426 160.261 2.994 18.094 0.725 0.361 0.680 0.185 0.339 1.610 0.981 0.448 0.706 0.190 0.637 1.098 0.755 0.690 1.529 0.461 0.822 1.387 165.472 3.060 18.481 0.741 0.369 0.697 0.189 0.342 2.408 1.002 0.461 0.722 0.196 0.651 1.133 0.772 0.709 1.545 0.470 0.841 1.419 165.594 3.200 19.296 0.775 0.386 0.734 0.198 0.350 4.091 1.048 0.489 0.755 0.209 0.681 1.206 0.807 0.750 1.579 0.487 0.881 1.485 165.837 2.940 19.390 0.669 0.333 0.745 0.178 0.287 1.421 0.918 0.386 0.661 0.211 0.659 1.216 0.822 0.759 1.628 0.498 0.896 1.512 151.636 2.984 18.040 0.679 0.338 0.682 0.176 0.312 1.463 0.932 0.411 0.671 0.191 0.606 1.101 0.758 0.692 1.535 0.463 0.825 1.392 165.389 2.984 18.040 0.679 0.338 0.682 0.176 0.312 1.463 0.932 0.411 0.671 0.191 0.606 1.101 0.758 0.692 1.535 0.463 0.825 1.392 165.389 3.006 17.349 0.684 0.340 0.648 0.178 0.329 1.485 0.939 0.424 0.676 0.180 0.610 1.040 0.723 0.656 1.486 0.444 0.787 1.328 173.276 3.027 17.462 0.689 0.343 0.654 0.179 0.330 1.727 0.945 0.428 0.681 0.182 0.615 1.050 0.728 0.662 1.490 0.446 0.792 1.337 173.318 3.250 18.706 0.740 0.368 0.710 0.192 0.342 4.400 1.015 0.471 0.731 0.201 0.660 1.162 0.782 0.724 1.541 0.473 0.853 1.439 173.742 3.015 21.119 0.889 0.444 0.806 0.216 0.382 2.905 1.175 0.550 0.834 0.232 0.764 1.336 0.882 0.827 1.692 0.529 0.964 1.625 142.767 International Educative Research Foundation and Publisher © 2020 pg. 267 Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 Vitamin-mineral supplements used in diets in three rearing stages (quantity / kg of product) included: pre Initial: vit. A 1,835,000 I.U. vit. D3 - 335,000 I.U. vit. E - 2,835 mg; vit. K3 - 417 mg; vit. B1 - 335 mg; vit. B2 - 1,000 mg; vit. B6 - 335 mg; vit. B12 - 2,500 mcg; folic acid - 135 mg; biotin - 17 mg; niacin - 6,670 mg; calcium pantothenate - 1,870 mg; Cu - 1,000 mg; Co - 35 mg; I - 170 mg; Fe - 8,335 mg; Mn - 10,835mg; Zn - 8,335 mg; Se - 35 mg; Choline Chloride 50% - 135,000 mg; Methionine - 267,000 mg; Coccidiostatic - 13,335 mg; Growth Promoter - 16,670 mg; Antioxidant - 2,000 mg. Termination: vit. A - 1,670,000 I.U. vit. D3 - 335,000 I.U. vit. E - 2,335 mg; vit. K3 - 400 mg; vit. B1 - 100 mg; vit. B2 - 800 mg; vit. B6 200 mg; vit. B12 - 2,000 mcg; folic acid - 67 mg; biotin - 7 mg; niacin - 5,670 mg; calcium pantothenate - 2,000 mg; Cu 2,000 mg; Co - 27 mg; I - 270 mg; Fe - 16,670 mg; Mn - 17,335 mg; Zn - 12,000 mg; Se - 70 mg; Choline Chloride 50% 100,000mg; Methionine - 235,000mg; Antioxidant - 2,000 mg. 3. RESULTS AND DISCUSSION Regarding the formulation principle (Linear and Nonlinear), the performance (Tables 3 and 4) was very similar in relation to the studied parameters. However, when simulated values of 50% below the historical average, performance was significantly impaired in this specific condition. If all essential nutrients are maintained in an adequate proportion to the energy density of the diet, body weight and feed conversion are favored by increasing the energy density of the feed. This condition makes it possible to apply models for maximum profit (nonlinear formulation), aiming to estimate the most appropriate proportion of weight gain according to the price paid by the market, producing quality carcasses. This worsening in live weight, weight gain, feed consumption and feed conversion is mainly due to the lower energy : nutrient content offered in this diet (-50%), which was inherent to the formulation principle ( nonlinear), which does not aim at the best broiler performance, but at the economic optimization of production. As for the profit margin (Table 5), it was demonstrated that the principle of nonlinear formulation allows to significantly reduce losses (P <0.05), mainly under unfavorable conditions in the market price of chicken. Table 3 - Live weight, weight gain, feed intake and feed conversion for female broilers, according to age and the linear model (LM) and nonlinear model (NLM) formulation principle. Trataments Live weight (kg) 1 - 21 days 1 - 42 days 1 - 56 days 1 - 21 days Weight gain (kg) 1 - 42 days 1 - 56 days Feed consuption (kg) 1 - 21 days 1 - 42 days 1 - 56 days Food conversion (kg/kg) 1 - 21 days 1 - 42 days 1 - 56 days Normal LM 0.93 a 2.71 a 3.81 a 0.89 a 2.7 a 3.8 a 1.3 a 4.8 b 7.4 b 1.4 b 1.8 c 2.0 c NLM+25% 0.94 a 2.63 ab 3.67 ab 0.89 a 2.6 ab 3.6 ab 1.3 a 4.9 ab 7.9 ab 1.4 b 1.9 ab 2.2 ab NLM+50% 0.93 a 2.63 ab 3.64 ab 0.88 a 2.6 ab 3.6 ab 1.3 a 4.9 ab 7.6 ab 1.4 b 1.9 b 2.1 b NLM-25% 0.88 bc 2.59 ab 3.60 b 0.84 bc 2.5 ab 3.6 b 1.3 a 4.9 ab 7.7 ab 1.5 b 1.9 b 2.2 b NLM-50% 0.85 c 2.56 b 3.60 b 0.80 c 2.5 b 3.6 b 1.3 a 5.0 a 8.0 a 1.6 a 2.0 a 2.3 a 0.91 ab 0.0004 2.82 2.61 ab 0.2437 3.16 3.63 ab 0.2524 3.54 0.87 ab 0.0004 0.69 2.6 ab 0.2437 3.22 3.6 ab 0.2524 3.58 1.3 a 0.3038 3.44 4.9 ab 0.3534 3.25 7.8 ab 0.2938 5.09 1.5 b 0.0027 4.87 1.9 ab 0.0024 2.95 2.2 ab 0.0010 3.42 Normal NLM P CV (%) a-b Mean values with same letter within a column are not significantly different (P<0.05); * kg of paid chicken (normal, + 25%, + 50%, -25% and -50%), according to the historical price from 2009 to 2010. International Educative Research Foundation and Publisher © 2020 pg. 268 International Journal for Innovation Education and Research www.ijier.net Vol:-08 No-12, 2020 Table 4 - Live weight, weight gain, feed intake and feed conversion for male broilers, according to age and the linear model (LM) and nonlinear model (NLM) formulation principle. Live weight (kg) 1 - 42 days 1 - 56 days Feed consuption (kg) 1 - 21 days 1 - 42 days 1 - 56 days Food conversion (kg/kg) 1 - 21 days 1 - 42 days 1 - 56 days 1 - 21 days Normal LM 1.03 ab 3.25 a 4.74 a 0.98 ab 3.20 a 4.69 a 1.4 ab 5.2 c 8.4 c 1.4 b 1.6 c 1.8 c NLM+25% 1.05 a 3.06 b 4.38 b 1.00 a 3.01 b 4.34 b 1.3 ab 5.3 bc 8.4 c 1.3 b 1.8 b 1.9 b NLM+50% 1.04 a 3.22 a 4.53 ab 0.99 a 3.18 a 4.48 ab 1.3 b 5.4 bc 8.7 bc 1.3 b 1.7 c 1.9 b NLM-25% 0.99 b 3.12 ab 4.55 ab 0.95 b 3.08 ab 4.50 ab 1.4 a 5.6 ab 8.9 bc 1.5 a 1.8 b 2.0 b NLM-50% 0.95 c 3.13 ab 4.60 ab 0.90 c 3.09 ab 4.56 ab 1.4 a 5.8 a 9.5 a 1.6 a 1.9 a 2.1 a 1.01 ab 0.0004 0.69 3.13 ab 0.0800 2.83 4.61 ab 0.1503 3.78 0.97 ab 0.0004 0.69 3.09 ab 0.0800 2.88 4.56 ab 0.1506 3.81 1.4 ab 0.1703 0.33 1.8 b <.0001 2.41 2.0 b 0.0002 3.40 Normal NLM P CV (%) a-c 1 - 21 days Weight gain (kg) 1 - 42 days 1 - 56 days Trataments 5.5 b 0.0022 3.19 9.0 b 0.0020 3.91 1.4 b 0.0002 4.86 Mean values with same letter within a column are not significantly different (P<0.05); * kg of paid chicken (normal, + 25%, + 50%, -25% and -50%), according to the historical price from 2009 to 2010 Table 5 - Absolute profit margin for female and male broilers, according to the relative price of the chicken and the principle of linear and nonlinear formulation. Profit margin (R$) Female Nonlinear Linear Nonlinear 1-42 days 1-42 days 1-56 days 3.68 a 3.68 a 4.80 a N +50% Linear 1-21 days 1.46 a N +25% 1.07 b 1.08 b 2.57 b 2.57 b 3.20 b 3.23 b 1.21 b 1.18 b 3.11 b 3.22 b 4.20 b 4.31 b Normal (N)¹ 0.64 c 0.69 c 1.50 c 1.46 c 1.75 c 1.67 c 0.74 c 0.76 c 1.94 c 1.89 c 2.58 c 2.36 c Relative price N -25% N -50% P CV (%) 0.30 d 0.31 d -0.07 e -0.07 e <.0001 7.82 0.49 d 0.35 d -0.56 e -0.76 e* <.0001 8.96 Linear 1-56 days 4.79 a 0.35 d 0.11 d* -1.11 e -1.46 e* <.0001 8.78 Nonlinear 1-21 days 1.63 a Profit margin (R$) Male Linear Nonlinear Linear 1-21 days 1-42 days 1-42 days 1.60 a 4.63 a 4.55 a Nonlinear 1-21 days 1.42 a 0.28 d 0.34 d -0.08 e -0.08 e <.0001 6.24 0.56 d 0.65 d -0.77 e* -0.53 e <.0001 6.22 Nonlinear 1-56 days 6.06 a Linear 1-56 days 6.25 a 0.70 d 0.42 d -1.20 e -1.52 e <.0001 10.49 Statistically different means (*) on the line by the T test (P<0.05); 1 Historical average price from 2009 to 2010 (kg of broiler paid to the producer); a-e Mean values with same letter within a column are not significantly different (P<0.05). Evaluating the EFE, IBE and BEC indices in the analysis of the bioeconomic profit margin (Tables 6 to 8). The data suggest that the bioeconomic energy conversion (BEC), proved to be more adequate to differentiate the evaluated formulation principles (Linear and Nonlinear), regardless of sex and period (Table 6). In relation to the bioeconomic indices evaluated (EFE, IBE and BEC / Tables 8 to 10), BEC differs by incorporating the most expensive item in a diet (energy), by measuring energy consumption according to bioeconomic conversion, that is, the best performance was analyzed in relation to the energy level of the diet. It follows that the lower the index, the better the cost/benefit ratio. Table 6 - Absolute Bioeconomic Energy Conversion (BEC) for female and male broilers, according to the relative price of the chicken and the principle of linear and nonlinear formulation. Relative price N +50% N +25% Normal (N)¹ N -25% N -50% P CV (%) Bioeconomic Energy Conversion (Female) Nonlinear Linear Nonlinear Linear Nonlinear Linear 1-21 days 1-21 days 1-42 days 1-42 days 1-56 days 1-56 days 1.22 e 1.17 e 1.41 e 1.47 e 1.51 e 1.61 e 1.43 d 1.40 d 1.71 d 1.76 d 1.87 d 1.93 d 1.84 c 1.76 c 2.1 c 2.20 c* 2.27 c 2.41 c* 2.26 b 2.34 b 2.69 b 2.93 b* 2.92 b 3.22 b* 3.37 a 3.51 a* 3.86 a 4.40 a* 4.15 a 4.83 a* <.0001 <.0001 <.0001 4.46 2.33 3.05 Bioeconomic Energy Conversion (Male) Nonlinear Linear Nonlinear Linear Nonlinear Linear 1-21 days 1-21 days 1-42 days 1-42 days 1-56 days 1-56 days 1.17 e 1.17 e 1.36 e 1.39 e 1.49 e 1.52 e 1.41 d 1.40 d 1.63 d 1.67 d 1.72 d 1.82 d 1.79 c 1.75 c 2.01 c 2.09 c 2.12 c 2.27 c* 2.43 b 2.34 b 2.63 b 2.79 b* 2.77 b 3.03 b* 3.36 a 3.51 a* 3.60 a 4.18 a* 3.97 a 4.55 a* <.0001 <.0001 <.0001 4.49 2.49 3.55 Statistically different means (*) on the line by the T test (P<0.05); 1 Relative price of the kg of the broiler paid to the producer. BEC =(total energy consumption×weighted feed cost/kg):(weight gain kg×live chicken cost); a-e Mean values with same letter within a column are not significantly different (P<0.05). International Educative Research Foundation and Publisher © 2020 pg. 269 Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 Through this strategy, and with the evolution from linear to nonlinear formulation, economic optimization by energy density becomes dependent, mainly, on the energy and protein prices of feed ingredients and the value of chicken/kg. This procedure, since it complies with the law of decreasing returns [2], admits through nonlinear programming the most adequate condition for energy density, which is not possible due to linear formulation [1] [13]. Therefore, to improve the energy density of a feed, it is necessary to use the nonlinear formulation. Among the indexes evaluated (BEC, IBE and EFE), IBE presented the highest variation coefficient, with values between 9.48 to 20.27, demonstrating a great instability (Table 7). For EFE, the values were intermediate for CV, with values ranging from 2.96 to 4.67% (Table 8). As for BEC, the CV varied from 2.33 to 4.49%, thus demonstrating greater reliability for the evaluation of the averages of the current formulation principles (Table 6). Table 7 - Absolute Bioeconomic Index (IBE) for female and male broilers, according to the relative price of the chicken and the principle of linear and nonlinear formulation. Bioeconomic Index (Female) Nonlinear Linear Nonlinear Nonlinear Linear N +50% 1-21 days 0.53 a 1-21 days 0.55 a 1-42 days 1.38 a N +25% 0.48 b 0.48 b 1.13 b Normal (N)¹ 0.34 c 0.38 c 0.78 c N -25% 0.20 d 0.21 d 0.26 d Relative price N -50% P CV (%) -0.14 e -0.13 e <.0001 10.76 Bioeconomic Index (Male) Nonlinear Linear Nonlinear Linear Nonlinear Linear 1-56 days 1.81 a 1-56 days 1.87 a 1-21 days 0.62 a 1-21 days 0.61 a 1-42 days 1.77 a 1.18 b 1.39 b 1.49 b 0.54 b 0.53 b 1.38 b 1.51 b* 1.87 b 0.81 c 0.88 c 0.92 c 0.40 c 0.42 c 1.01 c 1.09 c 1.33 c 1.35 c* 0.19 d 0.09 d -0.03 d 0.18 d 0.23 d* 0.36 d 0.38 d 0.32 d 0.24 d* 1-42 days 1.43 a -0.83 e -1.04 e * <.0001 12.61 -1.70 e -1.93 e * <.0001 20.27 -0.15 e -0.15 e <.0001 9.78 1-42 days 1.79 a -1.03 e * -0.85 e <.0001 9.48 1-56 days 2.30 a Linear 1-56 days 2.47 a 2.02 b -1.74 e -1.99 e * <.0001 18.92 Statistically different means (*) on the line by the T test (P<0.05); 1 Relative price of the kg of the broiler paid to the producer. IBE=weight gain – (A×CR), a being the ratio between the price of one kg of feed and the selling price of one kg of whole chicken (Guidoni, 1994; Meinerz et al., 2001); a-e Mean values with same letter within a column are not significantly different (P<0.05). Table 8 - Absolute Bioeconomic Efficiency (EFE) for female and male broilers, according to the relative price of the chicken and the principle of linear and nonlinear formulation. Relative price Nonlinear 1-21 days Bioeconomic Efficiency (Female) Linear Nonlinear Linear Nonlinear 1-21 days 1-42 days 1-42 days 1-56 days N +50% N +25% 2.53 a 2.16 b Normal (N)¹ 1.66 c 1.74 c 1.31 d 1.30 d 0.86 e 0.87 e <.0001 4.67 N -25% N -50% P CV (%) 2.61 a 2.17 b 2.28 a 1.88 b 2.20 a* 1.83 b 1.52 c 1.46 c 1.16 d 1.10 d 0.78 e 0.73 e <.0001 3.00 2.13 a 1.72 b Linear 1-56 days Nonlinear 1-21 days 2.02 a* 1.69 b 2.65 a 2.19 b 1.40 c 1.35 c 1.07 d 1.01 d* 0.72 e 0.67 e <.0001 2.96 Bioeconomic Efficiency (Male) Linear Nonlinear Linear Nonlinear 1-21 days 1-42 days 1-42 days 1-56 days 2.61 a 2.18 b 1.72 c 1.74 c 1.25 d 1.31 d 0.86 e 0.87 e <.0001 3.87 2.37 a 1.96 b 2.29 a* 1.91 b 1.58 c 1.53 c 1.19 d 1.15 d 0.82 e 0.76 e <.0001 2.53 2.17 a 1.86 b Linear 1-56 days 2.14 a 1.78 b* 1.50 c 1.42 c* 1.13 d 1.07 d 0.75 e 0.71 e <.0001 3.66 Statistically different means (*) on the line by the T test (P<0.05); 1 Relative price of the kg of the broiler paid to the producer. EFE = (weight gain income : feed cost) ); a-e Mean values with same letter within a column are not significantly different (P<0.05). International Educative Research Foundation and Publisher © 2020 pg. 270 International Journal for Innovation Education and Research www.ijier.net Vol:-08 No-12, 2020 According to the present experiment, it is evident that all the indexes evaluated (BEC, IBE and EFE) made it possible to measure the variations imposed on the normal market price (with ranges of 25 to 50%, for or less). In other words, what was already expected, due to the high magnitude imposed for price variation (increases or decreases of 25%). However, in relation to the main objective of the present proposal, regarding the comparison between formulation principles (linear and nonlinear), the differences were extremely distinct, evidencing very well that there was much more quality and sensitivity of measurement by the BEC index. Then, all indexes presented a significant (P) probability (P <0.0001). Despite this extremely favorable P, the different behavior between the different indices must be highlighted. While the EFE presented its values differentiated between the principles of formulation tending towards the higher relative prices, the IBE presented a trend towards the lower values of the relative price of the broiler. However, both rates were fluctuating. The BEC, on the other hand, showed a more consistent behavior, with the statistical significance of the differences between the averages associated with the lower ranges of relative price of the broiler, showing less oscillation of the trend and greater coherence of the index. It was observed that for both females and males, the amount of abdominal fat is related to the formulation principle, being significantly favorable (P <0.05) for nonlinear. Because there is a worse use of energy (deviated to fat deposition) for the principle of linear formulation (Tables 9 to 12). The average values for the absolute weight and the weight of the body components of the broilers, in grams, are presented in Tables 9 to 12. However, the body composition for abdominal fat, feet, head and neck, feathers and blood, were significantly affected (P <0.05) by the formulation principle adopted (Linear vs NonLinear). Table 9 - Average values for absolute weight (grams) of carcass and body components of female broilers at 42 days of slaughter, according with the linear model (LM) and nonlinear model (NLM) formulation principle. 42 days of age Carcass Abdominal fat weight Feet Head + neck Viscera Feathers Blood Normal LM 1930a 45ab 78.8a 141.3a 211.3a 105a 70a NLM +25% 1770a 61.3a 66.3ab 133.8ab 225a 97.5a 62.5a NLM +50% 1796a 45ab 62.5b 135ab 220a 115a 90a NLM -25% 1759a 47.5ab 66.3ab 126.3ab 198.8a 117.5a 65a NLM -50% 1895a 36.3b 66.3ab 123.8ab 211.3a 112.5a 63.8a Normal NLM P CV (%) 1785a 0.6350 9.45 41.3b 0.1697 27.48 60b 0.1600 14.36 120b 0.2780 10.45 208.8a 0.4224 8.43 106.3a 0.8060 20.28 63.3a 0.3882 28.65 Trataments a-b Mean values with same letter within a column are not significantly different (P<0.05). International Educative Research Foundation and Publisher © 2020 pg. 271 Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 Table 10 - Average values for absolute weight (grams) of carcass and body components of male broilers at 42 days of slaughter, according with the linear model (LM) and nonlinear model (NLM) formulation principle. Normal LM 2339 NLM +25% 2243a 36.3a 96.3a NLM +50% 2146a 33.8a NLM -25% 2345a NLM -50% Normal NLM P CV (%) a-b Carcass 42 days of age Feet Head + neck Abdominal a fat 41.3 weight Trataments a a a Viscera Feathers Blood 247.5 a 120 105a 162.5a 233.8a 155a 66.3b 91.3a 140a 232.5a 107.5a 107.5a 31.3a 98.8a 166.3a 256.3a 142.5a 105a 2270a 31.3a 97.5a 146.3a 263.8a 150a 77.5ab 2119a 0.6936 10.84 35a 0.9760 54.80 87.5a 0.6495 11.83 152.5a 0.6723 17.15 228.8a 0.5930 13.49 137.5a 0.3463 24.57 77.5ab 0.1285 28.51 98.8 163.8 a Mean values with same letter within a column are not significantly different (P<0.05). Thus, abdominal fat, when expressed in absolute value (g), was significantly reduced (P <0.05) for females by 56.29% (from 120.1 g to 67.6 g, respectively for the Normal LM and Normal NLM), at 56 days of age (Table 11). Table 11- Average values for absolute weight (grams) of carcass and body components of female broilers at 56 days of slaughter, according with the linear model (LM) and nonlinear model (NLM) formulation principle. Normal LM 2901a Abdominal fat120.1 weight a NLM +25% 2692a NLM +50% Trataments Carcass 56 days of age Feet Head + neck Viscera Feathers Blood 90a 217.5a 310a 185a 87.5a 98.3ab 82.5a 186.3ab 275a 180ab 90a 2749a 73.9bc 91.3a 187.5ab 253.8a 135b 92.5a NLM -25% 2673a 81.1bc 74.5a 166.3b 276.3a 157.5ab 97.5a NLM -50% 2673a 97.1abc 90a 182.5ab 305a 172.5ab 82.5a 160ab 0.3274 19.77 82.5a 0.8788 22.49 2723a 67.6c 82.5a 180ab 277.5a 0.3967 0.0116 0.7844 0.2696 0.4296 8.67 33.09 21.99 15.29 14.64 a-c Mean values with same letter within a column are not significantly different (P<0.05). Normal NLM P CV (%) International Educative Research Foundation and Publisher © 2020 pg. 272 International Journal for Innovation Education and Research www.ijier.net Vol:-08 No-12, 2020 Table 12- Average values for absolute weight (grams) of carcass and body components of male broilers at 56 days of slaughter, according with the linear model (LM) and nonlinear model (NLM) formulation principle. 56 days of age Abdominal Trataments Carcass Feet Head + neck Viscera Feathers fat weight Blood Normal LP 3455.5a 67.5a 135ab 211.3b 295a 190a 150a NLM +25% 3442a 58.4a 123.8b 217.5ab 321.3a 185a 135a NLM +50% 3551.5a 46.9a 135ab 226.3ab 336.3a 192.5a 127.5a NLM -25% 3494.8a 55.3a 132.5ab 207.5b 336.3a 187.5a 152.5a NLM -50% 3721.4a 63.4a 143.8a 270a 400a 212.5a 152.5a Normal NLM P CV (%) 3456.8a 0.4496 8.68 64.6a 0.5318 39.28 127.5ab 0.3233 9.27 223.8ab 0.2565 16.70 323.8a 0.5353 22.73 200a 0.9323 23.96 125a 0.8685 30.32 a-b Mean values with same letter within a column are not significantly different (P<0.05). There was a clear influence of the concentration of nutrients offered in normal price diets on body composition. In this way, it is directly related to the formulation principle adopted (Linear and NonLinear) and, also, the body composition is conditioned to variations in energy concentration : nutrients [9], inherent to the nonlinear principle, which because it is adopted by the spreadsheet PPFR, maintains energy density with adjustments concomitant with other nutrients [5]. The results also showed that the effects of the formulation principles were more characterized in females, mainly for the deposition of abdominal fat. Thus, the greater deposition of abdominal fat was already expected for females, due to their lower growth rate (genetic potential). Thus, excess energy is deposited as lipids in the body. From the above, it is evident the importance of studying mathematical models and new principles of formulation that integrate the current knowledge of the use and deposition of nutrients in the body tissues of the modern broiler, mainly in protein and fat, aiming at the optimization of its deposition in the housing [11]. And in this way, to produce better quality carcasses, for increasingly demanding customers, who want a lower fat content in the products consumed [12]. 4. CONCLUSION In this study, it was observed that the ration formulation, based on the nonlinear model, corrects the distortions of the traditional system (minimum / linear cost ration), resulting in an optimal solution in terms of the energy content of the diet. The nonlinear concept proves to be a great tool to be applied in diet formulations in order to increase the profitability of a broiler breeding. 5. ACKNOWLEDGMENT The authors would like to thank FAPESP, for the financial support, the technicians of the experimental International Educative Research Foundation and Publisher © 2020 pg. 273 Online-ISSN 2411-2933, Print-ISSN 2411-3123 December 2020 sector of Zootechnics and the Faculty of Veterinary Medicine of Araçatuba, in the assistance during the whole accomplishment of the experiment. Process : 2011/15664-6. 6. REFERENCES [1] AFROUZIYEH, M.; SHIVAZAD, M.; CHAMANI, M; DASHTI, G. Use of nonlinear programming to determine the economically optimal energy density in laying hens diet during phase 1. African Journal of Agricultural Research, v. 5, p. 2770 - 2777, 2010. [2] ALMQUIST, H. J. Interpretation of amino acid requirement data according to the law of diminishing returns. Arch. BIOCHEM. BIOPHYS. v.44, p.245-247, 1953. [3] ARAUJO, Raquel Bighetti et al. Modelos de superfície de resposta para predição do desempenho de frangos e elaboração de análise econômica. 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