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The objective of this study was to simulate total dry matter intake and cost of diets optimized by nonlinear programming to meet the nutritional requirements of dairy does and growing doelings. The mathematical model was programmed in a Microsoft Excel^{(r)} spreadsheet. Increasing values of body mass and average daily weight gain for growing doelings and increasing body mass values and milk yield for dairy does were used as inputs for optimizations. Three objective functions were considered: minimization of the dietary cost, dry matter intake maximization, and maximization of the efficiency of use of the ingested crude protein. To solve the proposed problems we used the Excel^{(r)} Solver^{(r)} algorithm. The Excel^{(r)} Solver^{(r)} was able to balance diets containing different objective functions and provided different spaces of feasible solutions. The best solutions are obtained by least-cost formulations; the other two objective functions, namely maximize dry matter intake and maximize crude protein use, do not produce favorable diets in terms of costs.

Feeding is one of the most important components of the livestock activity. The productive animal must be fed properly to express its genetic potential, and feeding represents a high proportion of the total production costs. In two small dairy goat production systems in North-western Rio de Janeiro State, Brazil, 41 to 73% of the total effective operating costs consisted of concentrates (

Nonlinear programming can be used to simulate scenarios from input data (

The Microsoft Excel^{(r)} spreadsheet was used to program a mathematical model that combines the conceptual and mathematical structures of the CNCPS - Cornell Net Carbohydrate and Protein System (^{2}

The diets for growing doelings and lactating does were formulated as a general nonlinear programming problem subjected to constraints of equalities and inequalities. Three different problems were optimized by considering three different objective functions separately:

The objective function Z (Eq. 1) is represented by the linear combination of constant c_{i}, i.e., the unitary dry matter cost of the i-th ingredient; x_{i} represents the unknown dry matter intake of the i-th ingredient. The objective function W (Eq. 2) is the total dry matter intake, and the objective function K (Eq. 3) is the proportion of the crude protein ingested (CPI) transformed into metabolizable protein; MEI and MPI are the intakes of metabolizable energy (MJ/day) and metabolizable protein (g/day) intakes, respectively; MEt is the metabolizable energy required (MJ/day); and MPt is the metabolizable protein required (g/day; _{max} corresponds to the maximum fiber retention capacity of the rumen (g/day); EFI is the effective fiber concentration of the diet (g/kg of dry matter); and FI_{j} is the fiber increment added to the minimum fiber content set (200 g/kg of dry matter). FI_{j }values were increased successively by adding 50 g/kg of dry matter constant increments to the minimum concentration of effective fiber for dairy does, and 25 g/kg of dry matter constant increments for growing doelings until feasible solutions were no longer achieved.

No. Equation 164 RFMmax = 8.5 × BW 165 MP _{g }= 0.290 × ADG, for growing animals 166 MFCP = 0.0267 × DMI 167 TPD = 0.88 × CPI 168 k_{m }= 0.35 × [AME ⁄ ∑ (x) × 0.001 ⁄ 18.8] 169 MEm = (315 + 31.5 × BW^{0.75)}/km, for mature animals 170 MP_{l−d} = 1.45 × MP × MPm × 1000, for mature animals 171 ME_{l−d} = (1.4694 + 0.4025 × FCM) × Milkyield × k_{l−d}, for mature animals 172 k _{l−d} = 0.624 173 MEm = 580 × BW^{0.75}, for growing animals 174 MEg = 23.1 × ADG, for growing animals 175 EUCP = 1.031 × BW^{0.75}176 MP_{m} = MFCP + EUCP + 0.2 × BW^{0.6}, for mature animals 177 MEg = 28.5 × ADG, for mature animals

All symbols and acronyms are based on definitions of

Constraints to the use of urea were also added. It is recommended that the urea supply should not exceed 40 g per 100 kg of body weight (BW), and two hypothetical situations were considered to balance rumen ammonia nitrogen (RANB, g/d):

RANB ≥ 0 (8)

or RANB ≥ −200 (9)

The RANB is a relationship between ammonia and carbohydrates available to the rumen microorganisms (

Simulations for growing doelings were made by varying the mass of the animal from 17 to 35 kg of BW with 3 kg BW increments. The diets were optimized to meet maintenance requirements and nutrient demands generated by daily weight gains ranging from 0 to 150 g/day, with 25 g/day constant increments. The simulations for lactating does were made by varying the weight of the animal from 50 to 80 kg with 5 kg BW increments, and milk production ranging from 2 to 9 kg/day with 0.5 kg/day increments.

We solved the presented problems by using the Excel^{(r)} Solver^{(r)} spreadsheet. This tool uses a generalized reduced gradient algorithm to optimize nonlinear problems (

The prices of the feed ingredients used in the model (

Feed R$⁄kg _{i} A _{i} B1 _{i} B2 _{i} C _{i} A' _{i} B1' _{i} B2' _{i} C' _{i} CP _{i} CF _{i} Ash _{i} NFC _{i} Fiber _{i} pef Cd N kd _{1i} kd _{2i} kd' _{1i} kd' _{2i} kd' _{3i} k _{ri} k ^{4} _{ei} λ ^{4} _{r} Elephant grass 0.13 1.1 92.5 8.1 2.3 80.0 7.0 445.9 254.1 104.0 27.0 82.0 87.0 700.0 1.00 0.75 2 1.35 0.10 2.50 0.30 0.04 0.10 0.04 0.20 Cottonseed meal 0.90 76.8 318.5 3.4 21.3 19.0 171.0 244.2 63.8 420.0 19.0 63.0 190.0 308.0 0.36 0.75 1 0.71 0.06 3.00 0.10 0.03 10000 0.05 10000 Soybean meal 1.20 47.4 431.4 47.9 5.3 29.4 264.6 93.0 5.0 532.0 11.0 65.0 294.0 98.0 0.34 0.75 1 2.87 0.09 3.00 0.45 0.06 10000 0.05 10000 Wheat bran 0.66 58.0 96.0 21.0 5.0 35.8 322.2 314.6 52.4 180.0 45.0 50.0 358.0 367.0 0.02 0.75 1 2.45 0.04 3.00 0.70 0.12 10000 0.03 10000 Coast-cross hay 0.31 7.7 27.5 39.3 10.1 18.6 36.4 627.6 145.6 84.5 17.5 69.8 55.0 773.2 1.00 0.75 2 0.52 0.01 2.50 0.30 0.05 0.20 0.05 0.40 Alfalfa hay 0.50 54.7 106.8 9.5 19.0 171.0 19.0 300.0 200.0 190.0 20.0 100.0 190.0 500.0 0.92 0.15 2 1.50 0.09 2.50 0.30 0.05 0.18 0.06 0.36 Ground sorghum 0.48 3.2 76.9 15.5 3.3 100.5 630.5 59.8 52.2 98.8 41.0 17.0 731.0 112.0 0.34 0.75 1 0.25 0.06 1.50 0.12 0.05 10000 0.06 10000 grain Soybean oil 2.70 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1000.0 0.0 0.0 0.0 0.00 0.80 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Maize silage 0.31 42.5 28.9 6.0 7.7 0.0 273.0 415.3 124.7 85.0 32.0 70.0 273.0 540.0 0.85 0.75 2 3.00 0.10 2.75 0.25 0.06 0.15 0.05 0.30 Ground corn 0.60 8.0 52.0 5.0 1.0 78.0 697.0 93.0 5.0 66.0 21.0 40.0 775.0 98.0 0.34 0.80 1 0.26 0.02 3.00 0.35 0.06 10000 0.06 10000 Soybean grain 1.00 26.9 248.2 44.1 22.8 117.0 149.0 0.1 134.9 342.0 179.0 78.0 266.0 135.0 1.00 0.75 1 0.25 0.03 3.00 0.45 0.06 10000 0.05 10000 Urea 1.45 2812.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2812.0 0.0 0.0 0.0 0.0 0.00 0.00 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Brazilian currency R$ 1.00 = US$ 0.64 in July, 2011; and R$ 1.00 = US$ 0.32 in July, 2015.

The subscript i denotes the i-th feed ingredient.

See

High values indicate that λ_{r }" k_{e}.

The Excel^{(r) }Solver^{(r)} was efficient to obtain feasible solutions to the proposed problems. Simulations with increments for daily gain and milk yield resulted in positive linear relationships between production levels and MEI, and production levels and MPI (^{1} ^{1c} ^{1d} ^{1f}

The optimization for maximum dry matter intake, i.e., objective function W or Eq. 2, resulted in more expensive diets for growing doelings in comparison with the other objective functions (^{3} ^{3} ^{3} ^{3} ^{3} ^{3}

Linear optimization systems require an estimate of the dry matter intake as an input to solve the problem of least-cost diets (

The metabolizable protein and metabolizable energy intakes increase as animal production increases, because of higher demands for nutrients generated by growth, milk yield, and pregnancy (^{1f}

Speculations are made about the advantage of maximizing the dry matter intake of farm animals.

The rumen microorganisms can synthesize protein from non-protein nitrogen and ammonia is the main source of nitrogen for microbial protein synthesis (

Dairy goat farming is an important activity that can generate income and wealth for farmers. This activity can produce enough wealth to the succession of the family business, which is an important tool for generating jobs and income (

The Microsoft Excel^{(r)} Solver^{(r)} allows for the balance of diets for dairy goats and growing doelings using different objective functions. Least-cost formulations provide better solutions in terms of overall costs of the diets than maximization of dry matter intake or crude protein use do. There is no net improvement of maximizing both dry matter intake and efficiency of use of crude protein. The predictions obtained with this model are in accordance with ruminant nutrition theories, and the nonlinear programming problem of the diet can be modeled to simulate different scenarios for decision-making, which is useful for developing strategies for increasing profitability of dairy goat production systems.

The fourth author is grateful for the postdoctoral fellowship provided by the Rio de Janeiro Research Foundation (Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro - FAPERJ), document no. E-26/101.429/2014, and the Brazilian Federal Agency for the Support and Evaluation of Graduate Education (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES), document no. E-45/2013-PAPDRJ.