Example Application in Ecological Management

Assume that you were interested in developing a conservation plan for a large landscape (100 000 ha or more) in the western United States. Prior to developing a plan of action, you divide the area into management units (50-100 ha each), and develop a set of alternatives for each management unit that describe the state of the unit over time, quantifying conservation values as well as economic value. Ecological value relates to the ability of an area or landscape to support the wildlife and fisheries populations of interest to a management organization. One unit of measure of ecological value might be the habitat units that are estimated for the landscape. These could be estimated for each piece of land in a landscape (i.e., each timber stand), then averaged to arrive at an average habitat unit for the entire landscape.

Assume also that there is a constraint that indicates a minimum revenue must be generated each year, and that revenue-generating activities must be dispersed across the landscape. While these and other constraints may limit what type of management can be assigned to the set of management units, the basic task is to assign an alternative to each unit such that the set of assignments leads to the highest conservation value. From prior testing of the solution generation process, you have decided that the initial temperature is 300 °C, and that the temperature will change by 0.995 with every adjustment to the solution, again using a slow-cooling process (this is the cooling rate). The simulated annealing search might begin with a randomly defined, feasible solution to the problem, with an ecological value of 1000 units.

In this solution, one of the possible alternatives is randomly assigned to each management unit, yet done in such a way that the constraints are not violated. From here, the following three iterations of the solution generation process might occur:

1. A random management unit is selected (e.g., number 123), and one of its alternative management actions is selected (number 5). The result of this adjustment is a feasible solution where the ecological value is 1060. Since the ecological value has increased, the process formally accepts this adjustment into the solution, and a new solution is created. The temperature is then changed (cooled) to 298.5 °C.

2. A random management unit is selected (number 645), and one of its alternative management actions is selected (number 3). The result of this adjustment is a feasible solution where the ecological value is 1003. Since the ecological value is lower than the best found so far, the process temporarily accepts this adjustment into the solution. The simulated annealing criterion is subsequently computed:

The result is the value 0.826. We then draw a uniformly distributed random number between 0 and 1 (0.153). Since the random number is lower than the simulated annealing criterion, we formally accept this adjustment to the solution, even though it results in a lower ecological value than what was found previously. In this case, at this temperature, there was only about a 17% chance that the solution generation process would reject the proposed inferior adjustment to the solution. As the temperature lowers (cools), adjustments of this magnitude will become less frequent. However, we want the solution generation process to have the ability to move across the solution space freely in the early stages of the process, in case the randomly defined initial solution is far from the optimal solution. The temperature is also changed (cooled) to 297.0 °C.

3. A random management unit is selected (number 58), and one of its alternative management actions is selected (number 11). The result of this adjustment is a feasible solution where the ecological value is 1055. Since the ecological value is lower than the best found so far, the process temporarily accepts this adjustment into the solution. The simulated annealing criterion is computed:

The result is the value 0.983. We then draw a random number between 0 and 1 (0.437). Since the random number is lower than the simulated annealing criterion, we formally accept this adjustment to the solution, even though it results in a lower ecological value than what was found previously. The process formally accepts this adjustment into the solution, and a new solution is created. The temperature is then changed (cooled) to 294.0 °C.

As the adjustments continue, perhaps to the 200th solution, we find that the temperature continues to decrease. In this case, it has decreased to 110.6 °C. At this point in the search, assume that the solution with the best ecological value is now 1659, and that this solution was generated at the 200th adjustment. Assume also that a random management unit is selected (number 364), and one of its alternative management actions is selected (number 2), and the result of this adjustment is a feasible solution where the ecological value is 1602, a decline in value of 57 units. This is the same difference in solution value as was noted between adjustments 1 and 2 (1060 - 1003). Since this solution value (1602) is lower (poorer) than the best solution value (1659), we calculate the simulated annealing criterion exp[ - (1659 -1602) /110.6]

and find that the result is the value 0.597. Thus there is only a 59.7% chance we make this adjustment, whereas previously (between adjustments 1 and 2) there was an 82.6% chance we made a similar adjustment to the solution.

The simulated annealing search process continues until the temperature is sufficiently small that the only adjustments that are allowed increase the solution value. For example, if the temperature was 10 ° C, there would only be a 0.3% chance that the adjustment above would be allowed.

Since adjustments to a solution are randomly selected, locating adjustments that only improve the value of the solution (1) might be costly, in terms of computer time; and (2) might be impossible to find, if there are no more improvements, suggesting a local optima has been located; assume that the search process has evolved into a simple greedy algorithm.

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