Diet formulation: making use of non-linear functions

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1 Diet formulation: making use of non-linear functions Rick Kleyn, Nutrition Consultant, SPESFEED (Pty) Ltd. Introduction Traditionally, feed formulation is achieved through the use of linear programming (LP). LP is an effective and powerful tool, but all the outcomes are based upon the assumption that the data is both linear and additive. There are numerous examples in animal nutrition where this is known not to be true. Format International (2013), have incorporated a non-linear methodology to solve what is known as the convex problem into their feed formulation system (Single-Mix ). This allows for complex two and three-dimensional models (or more, for that matter) to be added into a standard least-cost formulation program. Another shortcoming of LP is that it can t consider the inclusion of one nutrient or ingredient altering the digestibility of other nutrients. In this paper, some examples are shown of how non-linear functions have been included in a standard feed formulation system. As will be appreciated, the mathematics involved in a determination such as this, is complicated and a full description of the methodology used goes beyond the realms of this article. Non-linear models and feed formulation To create a non-linear function some form of modelling (non-linear) is required and in order to achieve a feasible solution, any non-linear functions used must be rather well-behaved in order to have a computationally efficient means of finding a solution (Bradley et al., 1977). Therefore, meaningful models for the nutrient(s) under consideration must be built. The freedom of definition of non-linear equations means that there is a danger that the user will expect a solution in an area that is either mathematically infeasible or unbounded. Diligence is required in order to overcome this problem. The primary requirement is that the equation should model the behaviour of the nutrient for the range of input values to be used. Other factors to consider are that equations used must be smooth and continuous and that the argument does not become negative. Importantly, both the input data, the inositol phosphate level of the ingredients in the matrix, for example, and some form of response data are required. Using these data, a regression model can be built (Excel does this well), and this can then be added to the feed formulation program. The model must be generated using the same data that is used for the matrix. From a nutritional perspective, it is helpful if the function goes through the origin (zero) in order to avoid false positives, for want of a better description. An example of a two-dimensional model It has long been believed that the addition of fat to poultry diets results in a so called extra caloric effect." This may well be true when small amounts of fat or oil are added to the diet, but the reality is that as the inclusion of oil into the diet increases, so its energy contribution decreases. In table 1, the estimated adjustment for fat inclusion is shown. It would be a simple matter to make an adjustment for age as suggested by Leeson and Summers, (2005). A reduction in energy realisation by younger birds as follows: <7 days of age 88% of expected energy is utilised; 7-28 days of age 95% of expected energy is utilised and older than 28 days all the energy from fat is available. In addition, it is known that as the fat level of the diet increases, pellet quality may decrease, bringing about an effective reduction in the energy level of the diet

2 Energy Correction (MJ/kg) (Teeter et al., 2012) and this could also be quantified. An example of a simple regression model for fat inclusion into a broiler diet is shown in figure 1. Table 1 Adjustment factors for the AME n of (added?) fat together with calculated and correction factors use in the model (Data from Leeson and Summers, 2005). Added Fat Inclusion Level (g/kg) Relative Energy Level (%) Calculated Energy Levels (MJ/kg) Energy Correction (MJ/kg) Figure 1: A model for correcting the energy contribution of added dietary fat to a broiler feed y = x x x R² = Added Dietary Fat (g/kg) Using LP it is impossible to reduce the energy contribution of an ingredient (in this case added fat) as its inclusion in a diet increases. The way the regression model derived above can be included into a standard feed formulation system is shown below. Note that for formulation purposes, the energy correction is the product of the energy correction factor (as determined by the model) and the physical amount of fat added to the diet. An adjustment is required to convert the result from g/kg to the percent inclusion in the diet, which is how LP is usually configured. The correction factor is then added to (or subtracted from) the ME constraint (the right-hand side) of the LP. Correction = (( f f f) X a) X f))/100 Where: Correction for fat inclusion is expressed in MJ/kg f = the g/kg of added fat. a = the age factor A model such as this needs to be validated, and to this end, a formulation exercise was carried out using a standard broiler grower diet. The control diet is a standard least cost diet formulated using Soya oil with

3 an energy level of 36 MJ/kg. Then, different exercises were carried out using the non-linear model (Table 2), exploring increasing levels of added fat, forced by constraint. This table warrants some explanation. As can be seen, the optimal cost of the non-linear model is lower than that of the standard LP solution. It is easy to understand that if you ascribe a higher energy value to added oil, then less of this expensive ingredient will be used and there will be a concomitant saving in cost. The moment an LP is artificially restricted by adding constraints, the cost of the solution increases. It can be seen, for example, that when less oil (5 kg/ton) or more oil (15 kg/ton) than the optimal inclusion is used, then the cost of the diet rises. As the inclusion of fat is forced upwards, the cost of the diet continues to escalate, which is how we would understand a feed formulation program to behave. In order to demonstrate that the mathematics of the non-linear solver was correct, an exercise was carried out whereby the amount of fat was fixed at 5 kg/ton, and then the specification of the diet was manually reduced by MJ/kg, as calculated by the equation above. The cost was identical but the ranging value was different. This would indicate that the non-linear method clearly accorded the fat a higher value when used at low levels, which is to have been expected. Table 2: The results achieved when validating the model using Format International Single-Mix Diet Cost ($/ton) Added Oil (kg/ton) Min Range Max Range Energy Correction (MJ/kg Feed) Least Cost - LP Non-Linear optimal kg/ton LP at 5 kg/ton kg/ton kg/ton kg/ton Standard LP optimisation. 2. Non-linear optimisation. 3. Standard LP but with the energy correction manually applied. 4. Constrained non-linear optimisation was carried out. The ranging values calculated by a feed formulation program indicate both the range in ingredient cost at which the formulation is stable, and the relative value of the ingredient in a particular diet. Comparing the ranging values of the standard formulation with those of the optimal non-linear solution would indicate that the fat has a higher value in the non-linear formulation. This is what was expected, as the energy value of the fat at a level of 8 kg/ton would be higher than the value of fat which had not had a correction factor applied. Clearly, if an ingredient is forced into a formulation, it can t have both a maximum and minimum ranging value, as can be seen from the table. When a constraint of 5kg per ton was applied (less than the optimum), the maximum ranging value was higher than the standard. Once the minimum value is exceeded, a negative ranging value indicates that even if the fat was free, no additional amounts would be used. In an attempt to estimate the impact of age on the use of fat in a diet, a series of broiler feeds was formulated making use of a correction for age. As can be seen from table 3 the adjustment for age made a small difference in the case of the Grower diet both in terms of cost and of the additional energy contributed by the added fat. In broiler production terms, the non-linear optimal solution should give the same result as the standard diet because in theory, they contain identical energy levels. This may not be entirely true as the de-facto energy content may well be lower. This is not the real issue though. At higher levels of fat inclusion, fat is not yielding as much energy as would be expected. The net result of this is that the true energy levels of high-fat diets are often overstated, and performance will be lower than expected.

4 Table 3: The effect of adjusting the ME of added fat as broilers age. Diet Cost ($/ton) Dietary energy (MJ/kg) Added Oil (kg/ton) Min Range Max Range Energy Correction (MJ/kg Feed) Starter (< 7 days) Grower (8-27 days) Grower - no correction Finisher (> 28 days) No correction for age was applied. A more complex example An aspect of diet formulation that is becoming increasingly intricate, is balancing calcium and phosphorous levels of broiler diets. This matter is complicated by the addition of phytase as demonstrated by Le tourneau-montminy et al., (2010) who could show how the enzyme impacts on both calcium and phosphorus nutrition. There is wide variation in the effect of added phytase in poultry diets, which contribute to unreliable diet formulation (Bougouin et al., 2014). The yield of phosphorus and other nutrients expected through the use of an exogenous phytase enzyme is multi-factorial. These would include: the specific catalytic and kinetic characteristics of the enzyme itself (dos Santos et al., 2014); the concentration of enzyme relative to the level of substrate, in this case myo-inositol hexakisphosphate (IP); the age of the bird; the calcium and/or limestone level of the diet and the calcium to phosphorus ratio. The level of IP in ingredients is variable both between different ingredients and between batches of the same ingredient. Variation in IP content of feed ingredients can be influenced by the cultivar grown, the rainfall, soil conditions and the rate of phosphorus fertilisation. Sunflower meal contains considerably more IP than soya bean meal; wheat contains 2.5 g/kg while maize contains 1.86 g/kg of IP. However, both grain sources contain variable quantities of IP, with CVs in excess of 13% (Tahir et al., 2012). An examination of a typical set of broiler diets revealed that as much as 15% more IP can be found in a Starter diet than in a Finisher diet. Cowieson (2014), points out that not all sources of IP are equally digestible, so an available IP value would be required in order for feed formulation to be accurate. Several organizations have developed calibrations for NIR analysis, which will allow us to determine the IP content of individual ingredients before they are incorporated into a model or diet. The mere inclusion of a phytase enzyme into a broiler diet changes the level of IP in that diet. This leads us to the classical dilemma we don t know what the IP level will be until we have included phytase, and we don t know what phytase level should be used until we know how much IP is in the diet. For normal nutrients, feed formulation programs are able to deal with this problem in a simultaneous manner, but where one nutrient impacts on the expression of another, this is more difficult. The value of a non-linear dose response typically published by the various phytase manufacturers is limiting, as they generally do not take the level of available dietary IP into consideration. Simple linear approximation can be used to mimic non-linear enzyme response and reasonable answers can be derived. This is a time consuming process, so nutritionists tend not to carry out the exercise on an ongoing basis. The use of calculators that approximate and deal with the non-linear nature of phytase addition are widely used, but these are not able to take the added dimension of variable IP content of the diet into consideration. The use of a true non-linear approach allows the nutritionist to formulate diets taking both the non-linear nature of phytase response and the level of dietary IP into consideration simultaneously. The phosphorus yield derived from the addition of a phytase to the diet is dependent on both the level of phytase to be used and the level of IP in the diet. In addition, Selle et al., (2007) and Bougouin, et al., (2014) were able to demonstrate that the lower the IP level of the diet (as measured by Non-Phytate Phosphorus),

5 the more efficacious phytase is likely to be. It has been reported that increasing the dietary calcium to phosphorus ratio (Amerah et al., 2014)) will reduce the efficacy of supplemental phytase, while simply increasing the level of Ca inhibits IP hydrolysis (Tamim and Angel, 2003). Until recently, nutritionists have not been able to adequately formulate diets using these relationships, so there has been no need for the suppliers of enzymes to present data of this nature. Rather, this relationship has had to be modelled for the purpose of this paper. The response curve published by DSM (2013) was used as a starting point for this exercise. Under practical conditions, it is unlikely that phytase will release more than 70% of the IP contained in a diet (Plumstead, 2014) so this value was set as the maximum yield to be expected of the phytase. By calculation, the percentage yields for the other levels of phytase inclusion were determined. The final yield of phosphorus was then derived by multiplying the percentage digestibility brought about by the phytase with the level of IP in the diet. In this example, it is assumed that the digestibility of IP is of a linear nature although this is unlikely. The model that was finally determined is: Phosphorus yield = ( p p p ) x i (r 2 = 0.934) Where: p = is the level of phytase in the diet in g/ton. i = the inclusion of IP in the diet in g/kg Figure 2: A model of expected phosphorus yield where both the level of dietary inositol phosphate and the level of phytase enzyme applied vary. As was done previously, this model was also validated by doing some practical examples. Although the Format system does allow for multiple non-linear relationships to be added to a formulation, it was decided to remove the relationship that calculates the energy contribution for phytase because it made it far more difficult to understand what was happening in each formulation. Thus, in the examples below, only digestible phosphorus and calcium have been considered.

6 The first challenge given to the model was to demonstrate how it would react to differences in the price ratio of inorganic phosphorus (MCP) to phytase (Ronozyme Hi-Phos). One would expect the amount of phytase used to decrease as its price relative to inorganic phosphorus increased. From table 4, it can be seen that this is indeed what happened. In the second exercise, the levels of IP in the diet were manipulated by forcing the inclusion of low IP-containing ingredients into the formulation. As can be seen from Table 5, the amount of phytase included in the formulation decreases as the levels of substrate are reduced. The standard feed used in this example is the same diet that was used in the first example as the 1:100 diet. This model is based on the assumption that all phytate is equally digestible, which may not be true, but which is a factor that could be considered when the data becomes available. Table 4: The impact of the inorganic phosphorus price on the inclusion of phytase in a typical broiler grower diet. Price Ratio MCP:Hi-Phos 1:1000 1:100 1:10 Cost/Ton Yellow Maize 7.5% Extruded Full Fat Soya 36% Soya O/C 46% Sunflower O/C 38 % Limestone Monocalcium Phos Salt Soya Oil HiPhos AME (MJ/kg) Crude Protein (g/kg) Fat (g/kg) Crude Fibre (g/kg) Calcium (g/kg) Total Phosphorus (g/kg) Digestible Phosphorus (g/kg) Sodium (g/kg) Available Lysine (g/kg) Available Methionine (g/kg) Dietary Inositol Phosphate (g/kg) Model phosphorus yield (g/kg) Phytase (g/ton) Adjusting nutrient digestibility and other examples Amerah et al., (2014) presented data which clearly demonstrates that the amino acid digestibility of a broiler diet is impacted upon by both phytase inclusion in the diet and the calcium to the available phosphorus ratio. Only a single level of phytase was used for this experiment (although a curvilinear response would be expected had more been used) making it a simple matter to add a relationship that adjusts amino acid digestibility (lysine in this case) to a changing calcium to phosphorus ratio.

7 Additional Lysine yielded (g/kg) = (( (0.0084*(Ca/AvlP)))x a) Where Ca = the dietary calcium level (g/kg) AvlP = the dietary available phosphorus level (g/kg) a = the average % increase in lysine digestibility when using phytase. Table 5: The impact of dietary inositol phosphate level on the inclusion of phytase in a typical broiler grower diet. Animal Standard No Sunflower Animal Protein Protein No Sunflower Cost Wheat Yellow Maize 7.5% Extruded Full Fat Soya 36% Soya O/C 46% Sunflower O/C 38 % Carcass Meal 55% Local Fish 65% Poultry By-Prod 50% Limestone Monocalcium Phos Salt Soya Oil DL Methionine L Threonine L Valine Lysine HCL Premix and Additives HiPhos AME (MJ/kg) Crude Protein (g/kg) Fat (g/kg) Crude Fibre (g/kg) Calcium (g/kg) Total Phosphorus (g/kg) Digestible Phosphorus (g/kg) Sodium (g/kg) Available Lysine (g/kg) Available Methionine (g/kg) Dietary Inositol Phosphate (g/kg) Model phosphorus yield (g/kg) Phytase (g/ton)

8 Adjusting nutrient digestibility and other examples Amerah et al., (2014) presented data which clearly demonstrates that the amino acid digestibility of a broiler diet is impacted upon by both phytase inclusion in the diet and the calcium to the available phosphorus ratio. Only a single level of phytase was used for this experiment (although a curvilinear response would be expected had more been used) making it a simple matter to add a relationship that adjusts amino acid digestibility (lysine in this case) to a changing calcium to phosphorus ratio. Additional Lysine yielded (g/kg) = (( (0.0084*(Ca/AvlP)))x a) Where Ca = the dietary calcium level (g/kg) AvlP = the dietary available phosphorus level (g/kg) a = the average % increase in lysine digestibility when using phytase. The use of the methodology described above is far ranging and powerful. For example, all of the examples described above could be combined into a single working model. The only limits to which parameters can be considered, will be the imagination of the nutritionist and the rigour of the equations used. Cowieson and Gous, (2014) explained that the effect of multiple enzyme effects should be considered on an individual nutrient basis rather than summed as a simple AME. Providing that these data exist, a model for inclusion into a feed formulation system could be built. Take-home message Mathematical models provide guidelines to managers for making effective decisions within the state of the current information, or in seeking further information if current knowledge is insufficient to reach a proper decision (Bradley et al., 1977). This means that they do not have to be correct," but rather they should be better than the approach they supersede. Nutritionists now have a flexible means of dealing with the nonlinear nature of nutrient inclusion in animal diets. This breakthrough will allow us to re-evaluate the way in which diets are currently formulated. Although a powerful tool, non-linear programming is not straight forward from either mathematical or nutritional perspectives, and care needs to be taken to ensure that the data used and the models that are built up from these data are reasonable. The old adage of if you can t do it with a pencil and paper, then you can t expect the computer to do it either, is applicable. Although use of the methodology may bring about some savings in feed cost, the real advantage will be that animal performance is more predictable. References Amerah, A. M., Plumstead, P. W., Barnard, L. P. and Kumar, A. (2014). Effect of calcium level and phytase addition on ileal phytate degradation and amino acid digestibility of broilers fed corn-based diets. Poultry Science 93 : Bougouin, A., Appuhamy, J.A.D.R.N.,Kebreab, E., Dijkstra, J., Kwakkel, R. P. and France, J., (2014). Effects of phytase supplementation on phosphorus retention in broilers and layers: A meta-analysis. Poultry Scienec 93: Bradley, S.P., Arnoldo, C. and Magnanti, T.L., (1977). Applied Mathematical Programming. Addison-Wesley Cowieson, A.J. and Gous, R.M., (2014). The expression of enzyme matrices by nutritional geometry. Proc Arkansas Nutrition Conference. Cowieson, A.J., (2014). Pers.comm.

9 dos Santos, T. T., Walk, C. L. and Srinongkote, S., Influence of phytate level on broiler performance and the efficacy of 2 microbial phytases from 0 to 21 days of age. J. Appl. Poult. Res. 23 : DSM (2013). Kaiseraugst, Switzerland Format International (2013). Woking, United Kingdom. Leeson, S. and Summers, J.D Commercial poultry nutrition (3rd edn). University Books, Guelph, Ontario, Canada. Le tourneau-montminy, M.P, Narcy, A. Lescoat, P., Bernier, J.G., Magnin, M., Pomar, C., Nys,.Y., Sauvant, D., and Jondreville, C., (2010). Meta-analysis of phosphorus utilisation by broilers receiving bcorn-soya bean meal diets: influence of dietary calcium and microbial phytase. Animal, 4:11, pp Plumstead, P.W., (2014). Pers.comm. Selle, P. H. and Ravindran, V. (2007). Microbial phytase in poultry nutrition. Animal Feed Science and Technology, 135: Selle, P.H. Cowieson, A.J., Cowieson, N.P., and Ravindran, V., (2012). Protein phytate interactions in pig and poultry nutrition: a reappraisal. Nutrition Research Reviews,25, pp 1?17 Tahir, M., Shim, M.Y., Ward, N.E., Smith, C., Foster, E., Guney. A.C. and Pesti, G. M. (2012). Phytate and other nutrient components of feed ingredients for poultry. Poult Sci : Tamim, N.M., Angel, R. and Christman, M. (2004). Influence of dietary calcium and phytase on phytate phosphorus hydrolysis in broiler chickens. Poult Sci 83: Teeter, R., Beker, A., Brown, C., Broussard, C., Newman, L. and Ward, N Production and managerial considerations influencing the calorific efficiency of growing broilers. Proceedings of the Arkansas Nutrition Conference.

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