Protecting us from wildfires and wildlife from us
How mathematical optimization can achieve both
The tragic year of 2020 began with continuing footage of Australian bushfires all over the news. The 2019–2020 bushfire season has been called the “Black Summer” in Australia; by end of May, fires had burned 46 million acres and killed at least 34 people. Later the same year, wildfires ravaged through much of the western United States. The magnitude of these fires is like none we’ve witnessed in decades, and unfortunately, we only expect them to get worse over time due to climate change.Thankfully, the risk of wildfires can be mitigated through some techniques called fuel management, including prescribed burning and mechanical clearing of trees.
Let’s take a deep dive into the 2017 paper written in Applied Mathematical Modeling by Ramya Rachmawatia, Melih Ozlena, John Hearnea, and Karin Reinkea:
“Fuel treatment planning: Fragmenting high fuel load areas while maintaining availability and connectivity of faunal habitat.”
This is how Rachmawatia et. al. describe strategies for mitigating wildfire risks, in their own words: “This is the process of altering the structure and amount of fuel accumulation in a landscape. Reducing the fuel load in the landscape… helps to prevent or minimize the spread and intensity of wildfire.” Specifically, the objective is to “fragment high fuel load areas as much as possible.”
Not only do the preventative treatments, prescribed burning and clearing, reduce the overall fuel available, but they also impact the local animal species and the overall ecosystem. This is especially clear when you consider that the survival of some animal species depend on particular trees, which may take up to a decade to regrow after they have been removed. Hence, we see the natural rise of a second objective: to reduce the impact on local ecosystems. Specifically, the authors aim to maintain not just the availability of animal habitats, but the connectivity of the habitats as well. Intuitively, a large and interconnected population is healthier than a population that is partitioned into many small, disconnected subpopulations.
This paper provides the first “multi-period fuel treatment model that takes into account habitat connectivity.” Specifically, it provides a solution which optimally schedules the treatments for any desired landscape! This model is designed as an integer program. That means that decisions are modeled as variables, some of which are binary (example: to treat or not to treat) and others can take on any integer value from 0, 1, 2, etc. (example: age of vegetation). Once combined into a formulation, including constraints and objectives, the program is solved to find an optimal solution!
In the model proposed by these authors, the single objective is the fragmentation of the high-risk fuel, and the most important constraint is the minimum requirement for the habitat connectivity. However, this single objective optimization problem tends to only have a single optimal solution. A single solution does not inform the decision maker of whether it is possible to improve upon the habitat connectivity from the bare minimum requirement. Often times, a multiobjective approach would show that with a small cost to the primary objective (high-risk fuel fragmentation), there could be significant gains in the secondary objective (habitat preservation). Therefore, a multiobjective optimization approach to this problem could be even more insightful to the decision makers!
For modeling the problem, imagine a landscape divided according to a simple square grid. Each block in the grid is assigned its own clock that gets reset to zero when that piece of the landscape is burned (either intentionally or naturally). In this way, the clock represents the amount of vegetation growth that has occurred since the last fire. There are a few relevant thresholds for these timers:
(1) General constraints on the frequency of fires per block, including the minimum and maximum interval between fires.
(2) Depending on the vegetation, there is a maturity threshold which indicates when the plants have grown enough to be suitable for the animals’ habitat. Mature blocks are desirable from the wildlife’s perspective!
(3) After a certain amount of time has passed, the block is considered to be high risk due to excessive vegetation accumulation. These blocks are undesirable from our perspective, because they lead to more dangerous wildfires!
Recall that both objectives relate to the connectivity of blocks in this grid. First, we want to minimize the number of high-risk blocks that are adjacent to one another. Second, we want to maximize the clusters of blocks with mature vegetation in them. Therefore, graph theory is greatly useful here! Each block is a vertex of your graph, which are then connected by edges, representing adjacency. These authors treat a block as a square with four neighbors, but it could also be adapted to include the “corner neighbors,” as well. Decision makers can choose the scale of how much area is represented by each block, as well as how many blocks to include in the overall grid. You can imagine that adding more blocks would increase the accuracy but also will slow down the optimization program. On the other hand, you could sacrifice some accuracy by using a reasonable number of blocks, each of which covers a large area of your landscape.
The computational results show that an optimization model with full scope of the landscape can quite quickly fragment the high-risk fuel structure. Sometimes, this is made more difficult when guaranteeing connectivity of habitats, and so the result includes a few clusters of high-risk blocks. Other times, the habitat connectivity constraint results in infeasibility, which is another reason that it is beneficial to solve the problem with this as an objectiverather than a constraint!
Optimization can be used to simultaneously protect humans and endangered animal species from trauma such as wildfires. As climate change presents increasingly high-stakes problems for the world to grapple with, it is essential for operations research and mathematical modeling to play a role creating solutions!
