Integrated Pest Management

Optimization

In this section:

Economics
Microeconomics
Optimization

Optimization

Until now we have discussed management decisions only a single management variable at a single moment in time. Real management situations are rarely that simple; We usually have to consider several management variables, all of which simultaneously affect crop yield–or we might have to make these decisions at several times throughout the season.

Optimization is a systematic procedure for finding the "best" solution (or solutions!) to a complex problem. It is necessary to state the objective explicitly, that is, to maximize something or to minimize something (e.g., to maximize yields, to maximize profits, to minimize effort, or to minimize costs). Generally we optimize only one objective at a time. For example, one can maximize profits or minimize costs, but not do both simultaneously. The solution is the combination of decision variables that optimizes the objective.

There are many different approaches to optimization, and we will touch on only three here: repeated simulation, linear programming, and dynamic programming.

Repeated Simulation

By repeated execution of a computer simulation model with different values for the input variables, we can explore the effects of different values of those variables on the final objective.

For example, suppose we have three partially resistant cultivars that require different levels of fungicide application to control a fungal disease. In this example, the more resistant varieties have either a lower quality or lower yield than the most susceptible one. Execute the simulation with a range of sprays (1—6) per season for each of the cultivars. Using the marginal analysis procedure, determine the profit for each set of input values.

Figure 1. Repeated simulation as a means of optimization. Each data point in this example represents the output from one execution of the simulator.

In this example, Cultivar C is the most susceptible and Cultivar A the least susceptible. Therefore, in the absence of fungicides, Cultivar A would yield the maximum profit. There are two optimum solutions if our objective is maximum profit: 3 sprays on Cultivar B and 4 sprays on Cultivar C.

The simulation approach is usually faster and cheaper than doing the optimization empirically in the field, but it's limited by how many runs of the simulation are feasible. In this example, the simulation had to be executed 20 times–not an unreasonable number for most simulators. But suppose that we had 5 cultivars, 4 different fungicides, 3 spray schedules, and 7 levels of fungicide, and that we wanted to look at the mean and variance of the profit using the past 10 seasons of weather data. The number of runs required would be

5 x 4 x 3 x 7 x 10 = 4200 runs

If each run cost $.50 on the supercomputer, the cost would be $2100. If each run took 1 minute on a microcomputer, it would take about 3 days of continuous computing.

Linear programming

Programming here refers to planning, not computer programming. Linear programming is used to solve problems of allocation (e.g., allocation of land area in a crop rotation plan, allocation of effort in a pest monitoring scheme, etc.).

The procedures are illustrated in the following simple example:

1. Define the objective function

Allocate weed control costs between herbicide application and hand weeding to maximize profit
In this example (for simplicity) we will make the yield and price constant and set the total revenue at $250/acre
Therefore, maximizing profit in this example means minimizing cost

2. Identify the constraints

Cost constraint: if x is the cost of hand weeding and y is the cost of herbicide application, then

X + Y <= 250

Weed control effectiveness constraint:

To achieve the yield that gives us the above total revenue ($250), we must invest at least $300/acre in hand weeding
The amount of hand weeding required can be reduced by $2 for every $1 spent on the herbicide
Therefore, if x is the cost of hand weeding and y is the cost of herbicide application, the constraint is given by

X + 2Y >= 300

Herbicide label constraint:

The amount of herbicide is limited because of possible phytotoxicity; the cost of the maximum allowable rate is $100/acre
Therefore the label constraint is given by:

Y <= 100

3.Determine the optimum solution

The optimal region of feasible solutions occurs where the constraint regions overlap.

Figure 2. Optimization by linear programming: A 2-dimensional allocation of herbicide application and hand weeding. Click on the image to add each constraint. The gray area represents the optimal region, given the constraints applied.

The optimum solution (given that the objective is to minimize cost) occurs where a line parallel to the cost constraint line (equal costs) is as far away from the cost constraint line as possible, while remaining within the optimal region. In this example it would be $100 invested in the herbicide and $100 invested in hand weeding.

Computer software exists for a wide range of linear programming applications. Linear programming can handle a large number of allocation variables (it is not limited to 2 dimensions as we are on a 2-dimensional graph), and it can handle huge numbers of constraint functions. The models must be linear, or they at least must be able to be approximated by linear functions. These models are not dynamic (that is, they cannot allocate variables through time), but they can approximate a dynamic solution by repeating the analysis at intervals through time. Linear programming cannot handle stochastic models, but probability distributions can be created by repeating the analysis with different constraints that vary according to known probability distributions.

Dynamic programming

Very commonly a pest manager has to make a series of decisions at different points in time throughout the season. One optimization technique that is particularly useful for solving sequential decision problems is called "dynamic programming."

As is illustrated in the following example, the optimum sequence of decisions is not simply a matter of making the optimum decision at every decision point. Very often the optimum sequence is counter-intuitive. To simplify the example, we will make just two decisions, with an interval of time between them, and examine the results of those decisions at some interval of time after they are made.

Suppose at each decision period we have three possible alternatives:

Do nothing; no cost; no insect mortality
Low dose of insecticide; costs $20/acre; kills 1/3 of insects
High dose of insecticide; costs $100/acre; kills 3/4 of insects

Further suppose that the total revenue accumulated during a time period is equal to $200 minus $1 times the pest population at the beginning of the time period. (Each insect does $1 worth of damage.)
Also suppose that the insect populations increase 3-fold during each time period.

If we start with a pest population of 72, the accumulated profits during the first period are as follows:

No insecticide: $200 - 72 - 0 = $128
Low dose: $200 - (2/3)72 - 20 = $132
High dose: $200 - (1/4)72 - 100 = $82

The pest populations at the beginning of time period 2 are:

No insecticide: 72 x 3 = 216
Low dose: (2/3)72 x 3 = 144
High dose: (1/4)72 x 3 = 54

Assume a decision about treatment (no insecticide; low dose; high dose) can be made at the outset and after period 1. The profits accumulated during time period 2 and the final insect populations are shown in the accompanying figure.

Figure 3. A simple sequential decision example with 2 decision points. Click on the image to see the net benefits for two different pathways. Making the optimum choice at each decision point yields a lower net benefit than taking the most costly option the first time and doing nothing the second.

Making the optimum decision at each decision point would dictate using the low dose of insecticide at each decision point for a total profit of $132 + 84 = $216. However, the optimum sequence would be to use the high dose for the first spray and nothing for the second: $82 + 146 = $228.

This example simply illustrates the need for a systematic optimization procedure. The number of possible decision combinations increases with the power of the number of decision points. For our trivial example, N = 3² = 9. If we had 5 control alternatives and 7 decision points, N = 5⁷ = 78125.

The dynamic programming algorithm does not analyze all possible combinations but selects possible sets of decisions according to certain rules. The dynamic programming technique can handle nonlinear models, stochastic models, and a large, but limited, number of decision variables. It can be used where the system can be adequately modeled with a relatively modest number of variables.