Field-to-Crop Assignment

LINEAR ASSIGNMENT PROBLEM

Proper crop-field matching can increase yields by 15–25% compared to arbitrary assignment, yet most farms still rely on tradition or intuition. Each season, a farm manager must decide which crop to plant in which field — a decision repeated across every field, every year. This is the Linear Assignment Problem — one of the few combinatorial optimization problems solvable to proven optimality in polynomial time.

Where This Decision Fits

Agriculture operational chain — the highlighted step is what this page optimizes

Soil AnalysisTest pH, nutrients, drainage

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Crop-Field AssignmentMatch crops to fields for max yield

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Seed ProcurementOrder seeds for selected crops

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Planting ScheduleSequence field operations

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Growing SeasonIrrigation, pest control

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Harvest & TransportCollect and deliver crops

The Problem

From farm fields to optimization theory

You have a set of farm fields, each with unique soil, drainage, and sunlight characteristics, and a set of crops, each with different agronomic needs. The constraint is that each field gets exactly one crop, and each crop is planted in exactly one field. The question is: which crop should go in which field to maximize total expected yield across the entire farm?

A farm has 6 distinct fields with different soil types, drainage, and sunlight exposure. 6 crop types have varying requirements. Assigning each crop to exactly one field to maximize total expected yield is the Linear Assignment Problem — solvable optimally in O(n³) by the Hungarian method.

	Agriculture Domain	Assignment Model
	Farm field	Agent (row)
	Crop type	Task (column)
	Expected yield (tons)	Cost/benefit c_ij
	One crop per field	One-to-one constraint
	Maximum total yield	min Σ c_iσ(i)

LAP — O(n³) via Hungarian method (Kuhn, 1955)

Try It Yourself

Edit the yield matrix, add/remove fields and crops, then find the optimal assignment

Yield Matrix (tons/acre)

6 Fields × 6 Crops · Click any cell to edit

Choose a Scenario

A diversified family farm with 6 fields of varying soil (loam, clay, sandy) and 6 common crops. The balanced mix of soil types means each crop has a clear best field, but greedy can still pick suboptimally.

Select Algorithm

The Algorithm

Hungarian Method (Kuhn-Munkres)

Row Reduction

Subtract the minimum value in each row from all elements in that row.

Column Reduction

Subtract the minimum value in each column from all elements in that column.

Cover Zeros

Cover all zeros using minimum number of horizontal and vertical lines. If lines = n, go to step 4.

Extract Assignment

Find a set of n independent zeros (one per row and column). These form the optimal assignment.

Real-World Complexity

Factors beyond the basic assignment model

Soil pH Levels

Different crops thrive at different pH ranges, affecting yield significantly.

Crop Rotation

Previous years’ crops affect soil nutrients, adding temporal constraints.

Water Availability

Field proximity to water sources affects irrigation costs and feasibility.

Microclimate Zones

Elevation, wind exposure, and frost pockets create field-specific conditions.

Market Demand

Commodity prices fluctuate, making yield value dependent on market conditions.

Labor Requirements

Some crops need more manual labor; field accessibility affects labor costs.

References

Key literature on assignment problems

Kuhn, H.W. (1955).

"The Hungarian method for the assignment problem."

Naval Research Logistics Quarterly, 2(1-2), 83–97.

Munkres, J. (1957).

"Algorithms for the assignment and transportation problems."

Journal of the Society for Industrial and Applied Mathematics, 5(1), 32–38.

Jonker, R. & Volgenant, A. (1987).

"A shortest augmenting path algorithm for dense and sparse linear assignment problems."

Computing, 38(4), 325–340.

Basso, B., Dumont, B., Cammarano, D., et al. (2013).

"Environmental and economic benefits of variable rate nitrogen fertilization in a nitrate vulnerable zone."

Science of The Total Environment, 545–546, 227–235.

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Data shown is illustrative. This is a simplified model for educational purposes.