Introduction

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 (^{Vieira et al., 2009}). Least-cost optimization procedures are used to find the most suited combination of foods that meets animal requirements (^{Agrawal and Heady, 1972}). Linear programming tools generally able to solve the problem of diet formulation, such as the Simplex method (^{Agrawal and Heady, 1972}; ^{Tedeschi et al., 2000}). Nevertheless, the complexity of the animal physiology and the interactions among the food eaten and digestive and metabolic processes that occur in the animal organism (^{Dijkstra et al., 2005}) demand the use of nonlinear programming to obtain more accurate diet formulations (^{Jardim et al., 2013}; ^{2015}).

Nonlinear programming can be used to simulate scenarios from input data (^{De Los Campos et al., 2013}). On the other hand, simulation studies can be used to predict specific virtual situations before making decisions or to improve our understanding of certain phenomena. From this perspective, the aim of this study was to simulate scenarios where three objective functions were optimized: least-cost of diets were minimized, dry matter intake of simulated diets were maximized, and ratios of metabolizable protein intake to crude protein intake were maximized. These problems were considered as general nonlinear programming problems, in which target performances and nutritional requirements of dairy does and growing doelings and usual dairy goat feeds were used as inputs.

Material and Methods

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 (^{Fox et al., 2004}) to estimate nutritive value of feeds, and the ^{NRC (2007)} equations to calculate nutrient requirement of growing doelings and lactating does. The steady-state pool size and digestibility of fiber in the ruminoreticulum were modeled according to ^{Vieira et al. (2008a},b). Acronyms and symbols used in equations that describe the system are listed in Tables 1 and ^{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; Table 3). The term RFM_{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.

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 (^{Russel et al., 1992}; ^{AFRC, 1993}; ^{Fox, 2004}). In addition, a maximum limit of 50 g/kg of diet dry matter for crude fat concentration was set for all simulations (^{NRC, 2001}).

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 (^{Lasdon et al., 1978}).

The prices of the feed ingredients used in the model (Table 4) were taken in December 2010, as current market prices in the northern and northwestern regions of Rio de Janeiro State. The nutritional composition of the feeds was obtained from tables contained in CNCPS (^{Sniffen et al., 1992}), Nutrient Requirements of Beef Cattle (^{NRC, 1996}), and Nutrient Requirements of Dairy Cattle (^{NRC, 2001}).

^{1} Brazilian currency R$ 1.00 = US$ 0.64 in July, 2011; and R$ 1.00 = US$ 0.32 in July, 2015.

^{2} The subscript i denotes the i-th feed ingredient.

^{3} See Table 1 for definitions of symbols and acronyms.

^{4} High values indicate that λ_{r} " k_{e}.

Results

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 (Figures 1a, ^{1}b, ^{1c} and ^{1d}). Sometimes, the space of feasible solutions differed remarkably. However, the increments in the fiber content of the diet caused an increased dry matter intake until a maximum point was achieved. Afterwards, a sharp decrease in the solution space was observed at higher fiber concentrations in the diet (Figures 1e and ^{1f}). The number of feasible solutions was higher for fiber contents ≤500 g/kg of diet dry matter for lactating does (Figure 1e). However, for growing doelings, only the level of 725 g/kg of fiber in the diet reduced the space of feasible solutions considerably (Figure 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 (Figure 2a). The maximization of the crude protein utilization or objective function K (Eq. 3, Figure 2b) produced diets with intermediate costs and, obviously, the minimum cost optimization (objective function Z, Eq. 1) was the most efficient objective function to minimize diet costs (Figure 2c). For all simulations, the increase in the production performance (milk yield or daily gain) increased diet costs (Figures 2 and ^{3}). The RANB constraints (RANB ≥ 0 and RANB ≥ −200) did not influence the cost of the diets (Figures 3a, ^{3}b, ^{3}c and ^{3}d). The space of feasible solutions was insensitive to the RANB constraint, and although the dietary cost increased with more challenging performance levels, the same solution space can be observed by comparing Figure 3a with ^{3}b for milk yield, and Figure 3c with ^{3}d for average daily gain.

Discussion

Linear optimization systems require an estimate of the dry matter intake as an input to solve the problem of least-cost diets (^{Tedeschi et al., 2000}). However, the nonlinear nature of diet formulation is characterized by the interdependence between animal requirements and the food consumed (^{Jardim et al., 2013}; ^{2015}).Therefore, the solution or the optimized diet and its expected dry matter intake influences the values of the components of the constraints. In the model proposed in this study, intake is an output of the nonlinear optimization procedure.

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 (^{NRC, 2007}). However, dry matter intake has a physical limit, imposed by the dietary fiber content, and the maximal capacity of fiber retention in the rumen (^{Mertens, 1994}; ^{Vieira et al., 2008b}). The rumen size limits animal capacity due to fill, and because fiber generally passes from the reticulorumen more slowly, it has a great filling effect because of the distension it causes in rumen walls (^{Allen, 1996}). The simulations with higher fiber content in the diet limits the space of feasible solutions (Figures 1e and ^{1f}). According to ^{Mertens (1987)}, higher milk productions constrain the fiber content in dairy cow diets, and this was observed here for dairy does diets (^{Gonçalves et al., 2001}).

Speculations are made about the advantage of maximizing the dry matter intake of farm animals. ^{Mertens (1987}) developed simple mathematical models that can be used to predict maximum intake. However, simulations in which maximum dry matter intake was set as the objective function (Eq. 2) resulted in more expensive diets (Figure 2). The feedstuffs used as protein sources are, generally, more expensive than energy sources and, for this reason, simulations used to maximize protein utilization efficiency were made (Eq. 3). Nevertheless, the cost of protein-optimized diets using Eq. 3 as the objective function resulted in intermediary dietary costs (Figure 3). Least-cost diet formulation (Eq. 1) was the most effective procedure to reduce the cost of the diet under the same dietary constraints (Figure 3).

The rumen microorganisms can synthesize protein from non-protein nitrogen and ammonia is the main source of nitrogen for microbial protein synthesis (^{Russel et al., 1992}). The amount of ruminal ammonia nitrogen can be estimated by the sum of endogenous nitrogen recycling with nitrogen originated from degradation of dietary protein and dietary non-protein nitrogen, and by discounting nitrogen retained by bacteria (^{Russel et al., 1992}; ^{Vieira et al., 2000a},^{c}). The RANB indicates if rumen ammonia nitrogen is adequate to meet bacterial requirements. A positive RANB is essential to maximize ruminal degradation of the feed (^{Leng, 1990}), and so, nitrogen deficiency decreases carbohydrate fermentation and the growth rate of fiber fermenting bacteria like *F. succinogenes* that become unable to ferment cellobiose if RANB < 0 (^{Maglione and Russel, 1997}). The scenario in which RANB < 0 occurs in tropical pastures, specifically in the dry season, when the forage nutritive value and availability are reduced remarkably (^{Vieira et al. 2000b},^{c}). The constraint expands the space for feasible solutions compared with RANB ≥ −200, which could result in cheaper diets, but the cost of the diets did not differ between these two constraints (Figure 3). The CNCPS fractionation scheme is a useful tool to estimate the nutritive availability of protein and carbohydrate fractions in feeds and has been used to estimate the ruminal availability of protein and carbohydrates of tropical feeds (^{Cabral et al., 2000}).

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 (^{Vieira et al., 2009}), mainly in the state of Rio de Janeiro, because of its unique goat milk production systems that favor the development of special products for specific markets (^{Santos Junior et al., 2008}). Therefore, the control of production costs is mandatory. In that sense, nutrition models would assist in the optimization of small ruminant production scenarios (^{Tedeschi et al., 2010}). Among all variables regarding nutrition of ruminants, passage rate estimates affect the utilization of fiber by small ruminants too; therefore, models based on the retention of fiber in the rumen are needed to properly formulate goat diets (^{Tedeschi et al., 2012}; ^{Regadas Filho et al., 2014a},^{b}; ^{Jardim et al., 2013}; ^{2015}). In this regard, the simulation of different scenarios could help in the decision-making process and to improve the understanding of the dynamics of goat nutrition and feeding.

Conclusions

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.