How to calculate the cheapest fertilizer blend
Fertilizer is one of the biggest inputs on a row-crop or hay farm, and the cheapest program is rarely built from the product with the lowest price per ton. This page walks through the method and shows what it produces for seven crop presets. It is arithmetic on example numbers, not an agronomic recommendation.
1. Put every product on the same footing
Urea is 46% N, so a $520 ton holds 920 lb of nitrogen: about $0.57 per pound. Anhydrous ammonia at 82% N and $700 is about $0.43. Do the same for P2O5 and K2O, and for sulfur.
2. Count what each product carries along
DAP is 18-46-0, so every pound of P2O5 you buy in it brings about 0.39 lb of nitrogen with it. That nitrogen is worth something, and it lowers the nitrogen you still need to buy.
3. Let the solver trade the products off
Each target is a linear inequality in the pounds of each product, so the cheapest program is a linear program. The optimizer solves it, then re-solves to tell you what one more pound of each nutrient really costs and the price at which a product you are not using would start to pay.
Worked example, with the starter products and example prices, 160 acres
| Crop (approximate targets) | Optimum per acre | Field total | Each nutrient bought separately, per acre | Saved per acre |
|---|---|---|---|---|
| Corn after soybeans (about 180 bu/ac) | $137.11 | $21,937 | $146.76 | $9.66 |
| Corn after corn | $157.59 | $25,214 | $167.78 | $10.19 |
| Soybeans | $75.48 | $12,077 | $75.48 | $0.00 |
| Winter wheat | $95.56 | $15,289 | $103.60 | $8.04 |
| Grass hay (3 tons/ac) | $131.91 | $21,106 | $141.82 | $9.91 |
| Alfalfa (established, 5 tons/ac) | $178.17 | $28,508 | $178.17 | $0.00 |
| Vegetable garden, per acre equivalent | $155.68 | $24,909 | $164.30 | $8.61 |
Computed by the same solver the test suite runs. The example prices are illustrative, not quotes, and the crop targets are rounded rules of thumb.
What to check before you buy
Set the targets from your soil test and your extension service, replace the example prices with your quotes, and remember what the tool does not model: application method and timing, nitrogen loss, salt and burn limits, lime, micronutrients, freight and application cost.
Common questions
How do you calculate the cheapest fertilizer blend?
Write down every product you can buy with its price per ton and its grade (N, P2O5, K2O and S as a percent of weight). Convert each to a price per pound of nutrient, remembering that DAP, MAP and ammonium sulfate carry more than one. Then find the pounds of each product per acre that meet your nutrient targets at the lowest total price. By hand you can price two or three nutrients one at a time; with seven or more products and four nutrients at once, a linear program does it exactly.
Is the cheapest price per pound of nitrogen the cheapest nitrogen?
Not always. A phosphate product also delivers nitrogen, so the nitrogen you still need to buy is lower than the target. The optimizer credits every nutrient a product carries, which is why its answer can differ from a plan that prices each nutrient separately.
What does the example show?
With the example prices, anhydrous ammonia supplies most of the nitrogen, a phosphate product supplies P2O5 plus some nitrogen, potash supplies K2O, and ammonium sulfate is used only where a sulfur target forces it. Buying each nutrient separately from the cheapest source per pound costs noticeably more because it pays twice for the nitrogen in the phosphate. Your prices will give different answers, which is the point of running it with your own quotes.
Open the optimizer — related: the beef cattle ration optimizer and the production mix optimizer.