LPSolve – Lower and Upper Prices
Hello, and welcome to this LPSolve Knowledge Base video. In this episode, we’ll look at the Lower Prices and Upper Prices shown in the Results window, and how they help us understand ingredient pricing and inclusion in a feasible solution.
When you set up a linear programming ration formulation and find a feasible solution, the Results window displays two columns on the far right of the matrix, just to the right of the green Percent column. These columns are labeled Lower Price and Upper Price.
These values provide important information about how ingredient prices affect the solution and how individual ingredients interact with the formulation.
In this example, we’re looking at a solution for a calf milk replacer mixture. You’ll see different lower and upper prices listed for each ingredient. Let’s walk through a few examples to understand what these numbers mean.
First, consider Dry Fat 760. Its cost is $4,077 per metric ton, and it is not used in the formulation, so its inclusion rate is zero. The lower price is $164.12. This means the price would need to fall below $164 per metric ton before this ingredient would enter the solution. Since the current price is well above that level, it’s very unlikely this ingredient would be used.
The upper price is shown as 1 × 10³⁰. This indicates there is effectively no upper limit, because the ingredient is already excluded from the solution. Increasing its price further will not change its inclusion — it’s already out.
You’ll see similar behavior for ingredients such as dried whey and skim milk powder. These ingredients are not used in the solution, and the lower price tells us the point at which each would begin to enter the formulation.
Now let’s look at soy protein concentrate, which is priced at $1,500 per metric ton and included at 1.07 percent. Its lower price is $1,323, and its upper price is $4,100. This tells us that as long as the price remains within that range, the inclusion level remains stable.
If we reduce the price to $1,300 per metric ton and rerun the formula, we see the inclusion rate increase to the maximum allowed level of 5 percent. This indicates the price has dropped below the lower price threshold, making the ingredient more attractive to the solver.
Next, consider wheat gluten, priced at $1,750 per metric ton and already included at its maximum of 5 percent. The lower price is shown as negative 1 × 10³⁰. Since the ingredient is already at its maximum inclusion, lowering the price further will not change the solution.
However, the upper price is $19.86. If the price rises above this value, the solver will begin to reduce the inclusion of wheat gluten in the formulation. Increasing the price beyond that threshold causes the ingredient to become less attractive.
These upper and lower prices provide valuable insight into when an ingredient becomes attractive to the formulation, and when it will be reduced or removed. This information can be used to compare current market prices or futures values to understand when ingredients are likely to enter or exit a ration.
It’s important to remember that each ingredient’s prices are evaluated independently. Changing the price of one ingredient causes the solver to recalculate all lower and upper prices for the solution.
That’s it for this episode. Thanks for watching, and be sure to check out other LPSolve Knowledge Base videos to learn how to get the most out of the program.
