LPSolve Demo – Milk Blend

Hello, and welcome to this LP Solve demonstration. In this video, we’ll walk through how to formulate a liquid milk blend using whole milk and milk-replacer–type ingredients to achieve a target solids and nutrient specification at the lowest possible cost.

Let’s start by looking at the ingredient matrix. Ingredients are listed in the rows, and nutrients are shown in the columns. In this example, we’re using whole milk and water as the base ingredients, along with a 20:20 milk replacer as one possible option. We’ve also included several ingredients commonly used in calf milk replacers, such as dry fat, whey, skim milk powder, soy protein concentrate, WPC-34, wheat gluten, and a small set of mineral and vitamin premixes.

Our goal here is to allow LP Solve to choose between using a commercial milk replacer or building a blend from individual ingredients, depending on which option is least expensive while still meeting our specifications. Ingredient prices are entered in the cost-per-metric-ton column, and we’ve set reasonable minimums and maximums for several ingredients. For example, soy protein concentrate and wheat gluten are each limited to a maximum of five percent of the formula. We’re also assuming that if we use individual ingredients rather than a finished milk replacer, we’ll include a calf milk replacer vitamin–mineral premix.

We’ll begin by setting whole milk at a fixed amount of 50 percent of the blend, and then allow LP Solve to formulate the remainder. The nutrient specifications are shown in the constraint rows. In this example, we want an average solids concentration of 15 percent, with a maximum of 17 percent. Protein and fat both have defined minimums and maximums, ash has an upper limit, and calcium, phosphorus, sodium, chloride, and magnesium are set as minimum requirements.

With these constraints in place, let’s try to solve the formulation. You’ll notice that LP Solve reports an infeasible solution. This tells us that, with whole milk restricted to 50 percent, the available ingredients cannot meet all of the nutrient specifications simultaneously. In other words, 50 percent whole milk simply isn’t sufficient to reach these targets.

To explore this further, we’ll remove the minimum constraint on whole milk and rerun the model. This time, LP Solve finds a feasible solution, with whole milk entering at about 80.6 percent of the formula. In this solution, we meet our solids target between 15 and 17 percent, protein at just under 25 percent, fat near 25 percent, and all mineral constraints.

The resulting blend is approximately 80 percent whole milk, 15 percent water, about 3 percent soy protein concentrate, 1.9 percent wheat gluten, and small amounts of vitamin and mineral premixes. Notably, none of the whey-based ingredients, skim milk powder, or the commercial milk replacer are used in this solution.

The total formula cost is about 489 dollars per metric ton. If we look at the shadow prices, we can see why some ingredients are excluded. For example, whey would need to be priced below about 300 dollars per metric ton to enter the formula, compared to its current price of roughly 900 dollars. Skim milk powder would need to be under about 780 dollars per metric ton, far below its current price of around 3,000 dollars. Under these conditions, neither ingredient is economically competitive.

To simplify the formulation, we can now fix whole milk at exactly 80 percent and rerun the model. This produces another feasible solution with very similar results: whole milk at 80 percent, water at about 14 percent, soy protein concentrate at 3 percent, wheat gluten at 2 percent, and the remaining balance coming from minerals and premixes. The final blend contains about 15.2 percent solids, 24 percent protein, 25 percent fat, and meets all mineral specifications. You’ll also notice small amounts of dicalcium phosphate, salt, and magnesium oxide entering the solution to satisfy calcium, sodium, and magnesium requirements, respectively.

Next, let’s see what happens if we restrict soy protein concentrate and wheat gluten to a maximum of only 0.1 percent each. When we do this, the model struggles to meet the 24 percent protein requirement using whole milk alone. In fact, without higher-protein ingredients such as soy protein concentrate or WPC, the maximum protein concentration we can realistically achieve with whole milk is closer to 20 to 22 percent. We could confirm this by lowering the protein minimum and rerunning the model, but the conclusion is clear: achieving higher protein levels requires additional protein sources.

This example highlights the strength of LP Solve. It allows us to evaluate trade-offs between whole milk and milk replacer ingredients, identify binding constraints, and understand which ingredients are economically and nutritionally necessary to meet our targets.

Thanks for watching this demonstration, and be sure to check back for additional LP Solve training videos.