An email from purchasing lands early on Monday: corn has gone from 9,000 to 10,350 TRY a tonne. That is a fifteen percent jump. The production plan is set, the bins are full and the trucks load this afternoon. Whoever sits at the formulation desk has two options. Keep the formula as it is and absorb the difference, or solve the formula again at the new price. This article explains why the second option is almost always the right one, and how the calculation behind it works.
A formula is really a question
Look at a feed formula and you usually see a list of percentages: so much corn, so much soybean meal, a little oil, limestone, phosphate, premix. To a formulation program, though, a formula is a question. With the ingredients I have, at these prices, what is the cheapest mix that meets the animal's nutrient requirements?
Least-cost formulation answers that question. You supply three things: the ingredient prices, their nutrient values and the rules the formula has to obey, which are the constraints. Out of the countless mixes that satisfy every constraint at once, OptiFormula finds the one with the lowest cost per tonne. To read the result, look at three things: the inclusion rate of each ingredient, the cost per tonne and which constraints sit exactly on their limits.
Getting the constraints right is half the job
In our experience, most problems in the field come from the constraints, not from the program. Nutrient limits describe what the animal needs: metabolizable energy, crude protein, digestible amino acids, calcium, available phosphorus and so on. Ingredient limits reflect technical and practical concerns. You do not want too much wheat in a broiler diet because of viscosity. The fiber in sunflower meal affects pellet quality and intake. Oil has a ceiling set by what the pelleting line can handle.
Ratios are a third kind of constraint. The calcium to available phosphorus ratio, or methionine plus cystine relative to lysine, cannot be expressed fully by single nutrient limits, so they are written as constraints of their own. Finally, all inclusion rates must add up to one hundred percent. That sounds trivial, but forget to set aside fixed items such as premix and salt and the formula ends up with too much or too little room.
Then there is the question of margins. Ingredient analyses are averages and change from load to load, so many nutritionists formulate critical nutrients slightly above requirement. That safety margin has a cost, and shadow prices, which we come to below, are how you see it.
A small example
The numbers below are purely illustrative. A real mill's ingredient values and prices will differ. The point is to show the logic of the calculation.
Take a broiler grower diet. The ingredients available are corn, wheat, 46 percent soybean meal, 36 percent sunflower meal, soybean oil, limestone, dicalcium phosphate and L-lysine. Vitamin and mineral premix plus salt are fixed at 0.8 percent in total. The targets are at least 3,100 kcal/kg metabolizable energy, at least 21 percent crude protein, at least 1.15 percent digestible lysine, calcium between 0.87 and 0.95 percent and at least 0.44 percent available phosphorus. Wheat may go up to 20 percent, sunflower meal up to 8 percent, oil up to 5 percent and L-lysine up to 0.4 percent.
Prices per tonne are 9,000 TRY for corn, 8,800 for wheat, 18,000 for soybean meal, 11,000 for sunflower meal, 55,000 for soybean oil, 1,500 for limestone, 30,000 for dicalcium phosphate and 60,000 for L-lysine. With these data OptiFormula returns roughly 57.3 percent corn, 35.7 percent soybean meal, 3.4 percent soybean oil, 1.0 percent limestone, 1.75 percent dicalcium phosphate and 0.14 percent L-lysine, at about 14,560 TRY a tonne including premix. At these prices neither wheat nor sunflower meal makes it into the formula.
Look closely and something else stands out. Energy, protein, lysine, the calcium minimum and available phosphorus all sit exactly on their limits. The program has not given a single nutrient more than required. That is what a least-cost formula usually looks like: a handful of constraints are binding and the rest have room to spare.
When corn goes up
Back to Monday morning. Corn is now 10,350 TRY. If the formula stays unchanged, the cost per tonne rises by corn's share of the increase, about 0.573 × 1,350 = 773 TRY. Solve again at the new price and the picture changes. Wheat comes in up to its 20 percent limit, corn drops to 38 percent, oil rises to 4.2 percent to close the energy gap, soybean meal falls slightly and a little more L-lysine covers the lysine. The new cost is about 15,240 TRY a tonne. The increase is 679 TRY instead of 773.
Ninety-four lira a tonne may look small. For a mill producing 10,000 tonnes a month, that is around 940,000 TRY from this one price move alone, and the nutrient content is the same. The birds eat an equivalent diet. Only the mix of ingredients has changed.
A word of caution, though. Wheat jumping straight to 20 percent is correct on paper, but it raises practical questions. Is your enzyme program set up for it? Is there enough wheat in the bins? What happens to pellet quality? The program will not ask these questions for you. It only takes them into account if you write them in as constraints. A good formulator questions the result and, when needed, adjusts the limits and solves again.
Effects that change with dose
For most ingredients the contribution grows with the inclusion rate. Double the corn and the energy from corn doubles. That holds well enough for most nutrients, but not for all. The contribution of phytase and NSP enzymes to nutrient values does not rise in step with the dose, and it depends on the rest of the diet. Rounding to the mixer batch size is another case. OptiFormula takes such effects into account as well. In practice this means defining an enzyme matrix that depends on dose and substrate, close to its real effect, rather than entering it as a fixed value.
When there is no solution
Sometimes OptiFormula finds no formula at all. That usually means the constraints contradict each other. Set a very high protein minimum and a low maximum on soybean meal, for example, and there may be no mix that satisfies both. The first thing to check is whichever limits you changed last. Another cause we see often is a mistyped analysis. A protein value entered as 4.6 instead of 46 can make the whole formula impossible.
More than a formula
The formula is not the only thing OptiFormula returns. It also calculates a shadow price for every constraint and every ingredient. In our example, raising the energy minimum by 1 kcal/kg adds about 9 TRY a tonne, and one point of crude protein costs about 435 TRY. Sunflower meal, which is not in the formula, would need to fall to about 10,100 TRY a tonne before it enters. This kind of information is very useful in purchasing talks and when reviewing nutrient targets. We cover it in detail in our next article.
Least-cost formulation is not something you do once and leave. The formula is solved again whenever prices move, new analyses arrive or production brings new constraints. Over a year, the difference between mills that do this routinely and those that do not shows up clearly in the cost per tonne.
