Crop Rotation

Crop rotation is the agricultural practice of growing a series of different crops in a set of areas, called plots, over time. Crop rotation has benefits that include improved soil structure and reduced dependence on pesticides, herbicides, and fertilizers.

This example demonstrates a means of finding optimal crop rotations for a set of crops to be planted in a set of plots. This is done by formulating the problem as a discrete quadratic model (DQM) that can be solved using the D-Wave DQM solver.

Usage

To run the demo, type:

python crop_rotation.py

The demo program outputs an optimal crop rotation for a problem instance defined by a problem file. By default, the demo program uses the file data/problem1.yaml as problem file.

To solve a different problem instance, type:

python crop_rotation.py --path <path to your problem file>

The program produces an illustration of the solution such as this:

The plot is saved to output.png.

Generating Problem Instances

To generate random problem instances for testing, type:

python problem_gen.py NROWS NCOLS

where NROWS and NCOLS provide the dimensions of a 2D plot grid. The random problem file is written to stdout.

Additional parameters are described by:

python problem_gen.py --help

Code Overview

We frame the problem of crop rotation as a constrained optimization problem with the objective of maximizing plot utilization. This problem definition is based on the formulation described in Santos et al. (2008).

The constraints of the problem are as follows:

Each crop may be planted in any plot, and may be grown zero or more times in a rotation. Each crop may be planted only in a certain season, defined as a range of time units. Each crop grows for a fixed amount of time before harvest. While a crop occupies a plot, no other crop may be planted in that plot.

A plot can have zero or more adjacent plots. Crops are members of crop families. Crops from the same family may not occupy adjacent plots at the same time. Crops from the same family may not occupy a single plot in consecutive time periods.

Problem Parameters

These are the parameters of the problem:

• M : number of time units in rotation
• L : number of plots
• N : number of crops
• N_f : number of crop families
• F_p : set of crops in crop family p
• S : set of pairs defining adjacent plots
• t_i : number of time units required between planting and harvest for crop i
• I_i : range of time [a, b] during which crop i can be planted

Variables

The problem has M * L discrete decision variables, x_j,k. Each has cases i ∈ {0, ..., N}, which indicate that crop i is planted in plot k at time j (zero signifies no crop is planted).

We can use binary variable x_i,j,k to indicate whether crop i (or no crop, where i is 0) is assigned to discrete variable x_j,k. We use these x_i,j,k variables in the rest of this document wherever they make the math easier to follow.

Occupation

When x_i,j,k is 1, crop i occupies plot k from time j to time j + t_i.

Objective

The objective for the problem can be written as:

The DQM solver solves minimization problems, so we change this to a minimization problem by negating the expression.

Constraints

Now we consider our constraints. The first constraint is that two crops cannot occupy the same space. Since our x_i,j,k variables indicate planting, not occupation, we sum the x_i,j,k over each crop's grow time t_i to represent occupation. This is actually a set of constraints; there are M * L of them. They can be written as:

Our second constraint is that two crops from the same family cannot be planted sequentially in the same plot. This is also a set of constraints; there are N_f * M * L of them. They can be written as:

The third and final constraint is that two crops from the same family cannot be planted at the same time in adjacent plots. This is also a set of constraints; there are N_f * M * |S| of them. They can be written as:

Wrap-around Periods

Crop rotations are cyclic, so the period following M is 1. The constraints of the problem take wrapping into account. This is done by substituting j - r + M for j - r in the constraint inequalities, whenever j - r < 1.

Here is an illustration of a wrapping solution, where M = 12:

DQM Formulation

To formulate the crop rotation problem as a DQM, we want to choose linear and quadratic biases so that the solution energy will be at a minimum when utilization is maximized and the problem constraints are satisfied.

The DQM solver solves unconstrained problems, so we use a penalty parameter to relate our constraints to our objective. We use a single parameter, gamma, as our constraints are equally important. We choose a value for gamma that is larger than the maximum change in the value of our objective that can be obtained by changing one variable. This ensures that our constraints will be satisfied. One such value is 1 + max(t_i).

Note that the DQM solver allows each variable to have a different number of cases, or possible values. To reduce the total number of biases in the problem, we restrict the cases of the x_j,k variables to the set of crops that can be planted in period j, as defined by I_i.

Linear Biases

The linear biases are the negated time values -t_i.

Quadratic Biases

The quadratic, or interaction, biases arise from the constraints. As all of our constraints are sums of variables that must equal either 0 or 1, we can use a simple penalty scheme for each constraint. That is, we add a positive number (our penalty parameter, gamma) to the quadratic bias for every unique pairing of the variables that appear in each constraint's sum.

References

Santos, L. M. R. dos et al. "Crop rotation scheduling with adjacency constraints." Springer Science+Business Media, LLC, 26 Nov. 2008.

License

Released under the Apache License 2.0. See LICENSE file.