Fertilizer Calculator: NPK Blend and Rate Planner

Nutrient mass balance

For product rates $x_j$ and label fractions $a_{ij}$, delivered nutrient rate is $y_i=\sum_j a_{ij}x_j$. All rates are calculated in kg/ha on an N–P₂O₅–K₂O basis. Label percentages are divided by 100. Field masses equal product rate times area in hectares. This follows Montana State Extension MT200703AG (July 2019), PDF pp. 2 and 7. Elemental P and K targets are divided by 0.44 and 0.83 respectively, following Iowa State Extension's garden guide (reviewed March 2026). The rounded extension factors are used consistently in both directions.

Custom grades retain individual 0–100% label bounds and must satisfy $N+0.44P_2O_5+0.83K_2O\le100.5\%$. The 0.5 percentage-point allowance is an author-selected margin for rounded labels and conversion factors, not a regulatory tolerance. This is a necessary elemental mass screen; the oxide-equivalent sum itself may exceed 100%.

Selection of two or three products

The optimisation is derived here. Let $t_i$ be the remaining target and $s_i=\max(t_i,1\ \mathrm{kg/ha})$. Closest match minimises $\sum_i |y_i-t_i|/s_i$, then cost, then shortfall, then total material rate. Meet targets first minimises $\sum_i\max(t_i-y_i,0)/s_i$, then cost, then total relative error, then total rate. Thus cost never buys an improvement in the first priority; a cost-first solution meets every target whenever feasible. Zero targets are penalised against 1 kg/ha in closest-match mode, but are not hard bans.

The piecewise-linear objectives change slope on the three nutrient target planes. Enumerating intersections of these planes, three zero-product-rate planes and the optional N allowance plane finds the best feasible vertex. Rates cannot be negative. Singular combinations are skipped; rows are scaled before solving. Relative tie tolerance is $10^{-9}$ and feasibility tolerance is $10^{-7}$ kg/ha. Reported signed differences and conservation residuals are rounded to $10^{-10}$ to suppress floating-point noise.

Cost, grade and doses

Cost of each product is its field mass divided by the quoted mass, multiplied by the entered price. Currency is a dimensionless accounting unit and is never converted. Blend grade is $100y_i/\sum_j x_j$. The first dose receives the selected fraction; the remaining fraction is shared equally among later doses. A single dose receives the full amount. Every dose has the same grade.

Triangle and nitrogen screening

Each grade is normalised by its N + P₂O₅ + K₂O sum and plotted in barycentric coordinates. The convex hull shows possible nutrient ratios before rate or N limits; the selected dot is not weighted by carrier mass. A missing product collapses the hull toward the remaining grades. Germany's separate annual organic N screen follows DüV §6(4); the mineral recipe is constrained only by the user's remaining N allowance.

• Mineral fertilizers with known label analyses and mass-based recommendations. Nutrient quantities are label nutrient supply, not a model of plant uptake, recovery or yield.
• Targets already subtract soil supply, manure credits, previous crops and earlier applications. The tool does not double-count those inputs.
• Each product may contribute zero. Selecting None or disabling product C removes its contribution. Custom percentages must each lie between 0 and 100, and converted elemental N+P+K must be no more than 100% plus a 0.5 percentage-point rounding allowance. Oxide-equivalent percentages are not elemental mass fractions. Passing this necessary check does not establish chemical compatibility or confirm a manufacturer's analysis.
• No stock limits, mixing minimums, carrier additions, losses, application charges, taxes or bag rounding. Prices are linear, in one user-selected currency for a common quote mass. The default 500/700/450 per 1,000 kg are illustrative assumptions as of 29 October 2026, not published prices.
• Field application is uniform. Split fractions are arithmetic allocations, not recommendations of date, placement or safe seed contact.
• Target-unit choices keep separate entries; the results always show both kg/ha and lb/acre. Area and mass use the viewer's unit conversion. The triangle always uses oxide nutrient ratios.
• Linear mass balance and vertex enumeration are derived and implemented here; agronomic conventions and reference numbers come from the cited extension guides.

Use for planning mineral fertilizer mass and cost, not for deriving crop recommendations. Very small label percentages or nearly identical grades can make a recipe numerically sensitive. Verify input analyses, weighing precision and every nutrient excess; a zero target is not enforced as a legal prohibition. Use products without the prohibited nutrient when a ban applies.

Germany, rules checked 29 October 2026: DüV §6(4) generally limits total N from organic and organo-mineral fertilizers including manure to 170 kg/ha/year averaged over the relevant holding area; it is not a blanket mineral N cap. The entered organic N must represent the entire relevant annual loading, including other plans. Compost's 510 kg/ha over three years, greenhouse distinctions, authorised exceptions and full holding balances are not implemented. The screen may therefore flag cases requiring a separate assessment. §13a(2) provides additional designated-area rules, including a generally 20% reduced aggregate N budget and a parcel-based organic N limit, with exemptions and Land provisions. The user must establish the remaining lawful mineral N allowance after those adjustments. This is a dated screening aid, not a legal compliance determination: check the regulation in force and local implementation.

The shaded triangle shows unconstrained nutrient-ratio reachability only. It does not establish plant availability, physical blend compatibility, segregation resistance or suitability for a crop. Manure, compost, controlled-release modelling, fertiliser chemistry, weather, closed periods, water setbacks and placement restrictions require separate assessment. The Mississippi reference-case source note explains the inconsistent MAP label and total in that example; the equations are not altered to reproduce those errors.