An Event-Response Tree-Based Resource Scheduling Method for Wildfire Fighting

Abstract

Dispatching firefighting resources effectively plays a vital role in wildfire management. To control the fire in a timely manner, resources should be dispatched in an effective and reasonable way. Moreover, the relationship between various resource-dispatching processes should be intuitive for firefighters to make decisions. In this paper, we propose a novel event-response tree-based model to dispatch different kinds of firefighting resources based on the fire suppression index (SI), which evaluates the effect of fire suppression by considering the time, cost, and effect of dispatching resources. To validate the proposed method, we compared it with the widely used mixed-integer programming (MIP) by using the historical fire data of Nanjing Laoshan National Forest Park. The results showed that the E-R tree-based resource scheduling can effectively schedule resources as well as the MIP model. Moreover, the relationship between various resource-dispatching processes in the proposed model is clear and intuitive for firefighters to make decisions.

1. Introduction

Wildfire is currently one of the most-severe natural disasters in the world and has been considered one of the eighth biggest natural disasters. Wildfires not only can cause vast amounts of damage, but also result in many people losing their lives [1]. In November 2018, a wildfire resulted in dozens of deaths in California, and more than 150,000 acres of land were burned [2]. Recently, wildfire has been an increasing trend in China [3,4]. If the decision-makers dispatch resources depending on their experience, this may lead to excessive or insufficient resource scheduling. Thus, it is necessary to improve the ability of decision-makers to dispatch firefighting resources in a more efficient way [5].

In practice, firefighting resource scheduling can be classified into single-resource scheduling and multi-resource scheduling [6] according to the number of resource scheduling objects. Single-resource scheduling refers to scheduling a single kind of fire resource [6]. Currently, approaches to scheduling firefighting resources focus on different issues. The risk-based model aims to obtain the best route for evacuees at each location [7]. Graph theory and network flow methods are used to optimize the route for firefighting resource dispatching [8,9].

Multi-resource dispatch refers to having multiple resources to dispatch. Some studies have explored the multi-resource scheduling model through a dynamic optimization model to minimize the number of firefighting resources dispatched [10]. For firefighting, a shorter rescue path means a shorter arrival time, so the shortest path is the factor mainly considered in most of the studies [11,12,13,14].

Vehicles were one of the primary resources to dispatch in previous research [15,16]. With the development of fire detection [17,18] and firefighting technologies [19,20], different kinds of resources can be dispatched. In the multi-resource scheduling method, there are multiple objectives that should be satisfied. Some studies proposed fuzzy multi-objective programming combined with a genetic algorithm to optimize the location of the fire stations [21]. They appropriately converted the original fuzzy multiple objectives into a single unified goal based on the evaluation of experts [22]. Since cost is a factor that constrains the number of resources that can be dispatched, some studies have proposed a multi-resource scheduling method that combines the features of fire, labor, and cost to schedule firefighting resources. In order to schedule resources more effectively, some methods that dispatch resources combined the fire-spread model [23] or combined with GIS [24,25] have been proposed. It is difficult to measure the input parameters required by the model due to the harmful nature and instability of fire. Thus, its computational cost makes it not easy to operate and maintain.

Variables are in the form of integers, for example a variable whose values are restricted to 0 or 1, indicating whether or not some action is taken. The integrality constraints allow the models constructed by MIP to capture the discrete nature of some decisions. Thus, the resource scheduling problems are most commonly solved by MIP [26,27]. Although MIP can obtain effective scheduling schemes, its calculation is not easy, and the relationship between various resource-dispatching processes is not intuitive. It may not be easy for firefighters to make decisions in a timely manner.

This paper proposed a new novel event-response tree (E-R tree) model to schedule firefighting resources. The tree-based model is constructed based on the fire suppression index (SI). SI is proposed to evaluate the fire suppression ability of firefighting resources, which considers the time, cost, and effect of dispatching resources. In order to evaluate the proposed method, we compared it with MIP using actual data. The evaluation results demonstrated that our approach provides an effective firefighting-resource-dispatching method in an intuitive way. In this research, our aims were to propose a tree-based firefighting-resource-scheduling model to visualize the process of resource scheduling, which can help decision-makers make rapid resource scheduling decisions when a fire breaks out.

2. Materials and Methods

2.1. System Abstraction

Usually, wildfire resource scheduling can be abstracted as follows. There are n fire stations, $M_1,M_2,M_3,\cdots ,M_n$, in the study area. There exist different kinds of resources that can be allocated by each fire station defined as $X_1,X_2,X_3,\cdots ,X_m$. In each fire station, there exists a number of scheduling methods denoted as $\phi =|{\phi }_1,{\phi }_2,\cdots ,{\phi }_m|$. The set of all scheduling paths is $\mathsf{\Omega }$. The time spent on resource scheduling is $T\left(\phi \right)$; the cost spent on resource scheduling and fire suppression is $C\left(\phi \right)$; the number of the dispatched firefighting resources is $N\left(\phi \right)$. An optimal scheduling scheme should schedule firefighting resources to meet the fire control requirement with the following constraints Equation (1):

$$\left\{\begin{array}{c}min\left(T\right(\phi \left)\right)\\ min\left(C\right(\phi \left)\right)\\ min\left(N\right(\phi \left)\right)\\ st.\in \mathsf{\Omega }\end{array}\right.$$

(1)

The above constraints are solved by MIP. In this paper, we propose a new E-R tree-based method to solve it.

2.2. E-R Tree

The event-response tree (E-R tree) enumerates and specifies the scheduling method used by the decision-maker. The construction process of the E-R tree is simple and intuitive, as shown in Figure 1. There are three different kinds of nodes in the E-R tree. The path from the event node to the root node is called the scheduling path, which represents all available scheduling processes:

Root: 

The root node is the topmost node in the E-R tree data structure. It represents the target of resource scheduling.

Response: 

The response node is the non-leaf node in the tree data structure, which is in the middle of the E-R tree data structure; it has more than 0 children, and it can be seen as a sub-step to reach the root node. The response node is composed of multiple event nodes or response nodes.

Event: 

The event node is the leaf node in the tree data structure, which is the bottom node in the E-R tree data structure. It consists of indivisible resources, which do not have children.

There are two kinds of relationships, “AND” and “OR”, between sibling nodes, as shown in Figure 2; “OR” indicates as long as any branch under the parent node is completed, while “AND” indicates the various branches under the parent node that must be completed.

2.3. Mixed-Integer Programming

The mixed-integer programming (MIP) model is usually used to optimally select the resources for forest fire extinction in the planning period [28]. The objective function of the model Equation (2) is to minimize the cost during resource dispatching. The objective function consists of two parts. The first part is to minimize the cost caused by dispatching the firefighting resources from the station to the fire field (consequently minimizing the arrival time, that is the shorter distance, the less the cost is). The second part is to minimize the selection cost of the resources, that is the fixed cost caused by choosing and using the firefighting resource (thus minimizing the associated cost of the number of resources, consequently minimizing the number of resources required).

$$min\sum _{i\in \mathcal{I},t\in \mathcal{T}}{C}_i·u_{it}+\sum _{i\in \mathcal{I}}{F}_i·z_i$$

(2)

These are subject to the following constraints:

$$\sum _{t\in \mathcal{T}}PE{R}_t·y_{t-1}\le \sum _{i\in \mathcal{I},t\in \mathcal{T}}P{R}_{it}·r_{it}$$

(3)

$$\forall t\in \mathcal{T},P·y_t\ge \sum _{t^{\prime }\in {\mathcal{T}}^t}PE{R}_{t^{\prime }}·y_{t-1}-\sum _{i\in \mathcal{I},t^{\prime }\in {\mathcal{T}}^t}P{R}_{i{t}^{\prime }}·r_{i{t}^{\prime }}$$

(4)

$$\forall i\in \mathcal{I},t\in \mathcal{T},A_i·r_{it}\le \sum _{t^{\prime }\in {\mathcal{T}}^t}{tr}_{i{t}^{\prime }}$$

(5)

Equation (3) denotes that the area of the fire suppression that firefighting resources can bring is greater than the wildfire area. This constraint can ensure that the objective function has a solution. In Equation (4), $y_t$ is a binary variable, when $y_t=0$, which means that the fire can be controlled in time period $t\in \mathcal{T}$. Equation (5) denotes that the firefighting resources spend time in transit before they can be used to control the fire. This constraint considers the scheduling time of firefighting resources, which is in line with the actual situation.

Table 1. Indices, sets, parameters, and decision variables in the MIP model.

Indices
t	Current time period.
i	Kind of firefighting resource.
Sets
$\mathcal{T}$	Set of time periods in fire suppression.
$\mathcal{I}$	Set of fire firefighting resource in fire stations.
Parameters
$C_i$	Cost per time period for resource $i\in \mathcal{I}$.
$F_i$	Fixed cost for resource $i\in \mathcal{I}$
$PE{R}_t$	Increment of fire perimeter in time period $t\in \mathcal{T}$.
$P{R}_{it}$	Performance of firefighting resource $i\in \mathcal{I}$ in time period $t\in \mathcal{T}$
P	Positive constant, large enough to establish constraints
$A_i$	Number of time periods required to arrive at the fire point by firefighting resource $i\in \mathcal{I}$
Decision Variables
$u_{it}$	Resource $i\in \mathcal{I}$ has been dispatched in time period $t\in \mathcal{T}$
$z_i$	Resource $i\in \mathcal{I}$ has been selected
$y_t$	If $y_t=0$, it means the fire has been suppressed successfully in time period $t\in \mathcal{T}$; we take $y_0$ as 1, which means the fire is not contained in the initial time period
$r_{it}$	Resource $i\in \mathcal{I}$ is putting out the fire
$t{r}_{it}$	Resource $i\in \mathcal{I}$ is on its way to the fire point in time period $t\in \mathcal{T}$

lists the symbols in the MIP model.

2.4. Fire Suppression Index

In order to evaluate whether the scheduled resources can satisfy the number of resources required to control the fire, we propose the fire suppression index ($SI$). It can be calculated with Equation (6). $SI$ indicates the effect of the fire suppression ability of the firefighting resources. The larger its value, the more obvious the effect is. If there are n resources dispatched, $SI$ is the summation of all the n resources. In order to control the fire, its value should be larger than the $SI$ value required to control the fire.

$$SI=W\times U=\left[\begin{array}{ccc}{W}_{\mathrm{time}}\hfill & W_{\mathrm{cost}}\hfill & W_{\mathrm{eff}}\hfill \end{array}\right]\times \left[\begin{array}{c}U\left(tim{e}_i\right)\\ U\left({cost}_i\right)\\ U\left(ef{f}_i\right)\end{array}\right]$$

(6)

where W is the attribute weight vector. It can be calculated through AHP (more details in Section 3.2.2). U consists of ${time}_i$, ${cost}_i$, and ${eff}_i$. ${time}_i$ is the time for resource i to reach the fire point; ${cost}_i$ is the cost spent on scheduling resource i; ${eff}_i$ indicates the effect that the firefighting resource i can have; W satisfies $W\times E=1$ (E is the unit array).

$$U\left(x\right)=\frac{c}{x}$$

(7)

where x indicates the corresponding attributes’ rating, which is classified into five levels (

Table 2. Grading standard table.

x	Arrival Time (M)	Cost (K)	Effect
5	>120	>10	Not obvious
4	90–120	5–10	Little obvious
3	60–90	2–5	Obvious
2	30–60	1–2	Very obvious
1	<30	<1	Extremely obvious

) based on the travel time criterion ($TTC$) [29] and firefighting distance criterion ($FFDC$) [30] for firefighting resource dispatch. c is a constant, which takes the value of 1 for the convenience of calculation.

3. Construction of the E-R Tree-Based Wildfire-Resource-Scheduling Model

The overall construction process of the E-R tree modeling is shown in Figure 3, which consists of four parts: data input, modeling, quantification, and scheduling path.

3.1. Data Input

In this study, the road network of the area can be obtained through Open Street Map (OSM). The locations of the fire stations around the fire point were obtained through Google Maps. The path length between the fire station and the fire point was calculated with a digital elevation model (DEM), which can be obtained through the geospatial data cloud (

Table 3. Data source.

Data	Description	Source	Resolution
ASTGTMv003	DEM	https://lpdaac.usgs.gov accessed on 20 July 2022	30 m
OSM	Road network	https://www.openstreetmap.org accessed on 20 August 2022	m

).

The number of firefighting resources to be dispatched varies for different scales of fire. In this study, we used historical fire data to evaluate the scale of the fire. For the areas without fire history data, the fire model can be used to estimate the fire scale. Generally, factors such as the wind speed, temperature, humidity, fuel type, and slope are used as the variables of the model. The fire model evaluates the corresponding fire spread speed, V, calculates the time T required for firefighting resource scheduling according to the distance L between the point and the rescue site, and then, estimates the area of the forest fire according to V and T [31,32]. Based on the fire area, the amount of firefighting resource required can be obtained.

3.2. Modeling and Quantification

3.2.1. E-R Tree Construction

The E-R tree enumerates and specifies all the possible scheduling paths that can be used to dispatch firefighting resources. The root node is the target of resource scheduling. Generally, it is to put out the fire. Under the root node, these response nodes are the sub-targets that should be completed to achieve the final target. The leaf node of the E-R tree is the event node, which is the resources. We can construct E-R trees based on the available resources in different fire stations.

3.2.2. SI of E-R Tree

In order to calculate the $SI$ of the event node with Equation (6), AHP [33,34] was used to determine the weights of different attributes. When the weights are estimated, the experts compare the corresponding two variables with each other. In the process of comparison, the experts score the different degrees of importance between the two sets of variables with

Table 4. Scale table of the 1 to 9 scale method.

Intensity of Scale	Explanation
1	$a_i$ and $a_j$ are equally preferred
3	$a_i$ is moderately preferred
5	$a_i$ is strongly preferred
7	$a_i$ is very strongly preferred
9	$a_i$ is extremely preferred
2, 4, 6, 8	The intermediate value of the adjacent grades above the
reciprocal	Contrary to the meaning of the above comparison

.

The judgment matrix represents the result of a two-by-two comparison between attributes, which can be obtained through expert evaluation. After normalization of the matrix, the eigenvectors $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)$ can be obtained with Equation (8), where ${\alpha }_i$ denotes the weight value of the $i^{th}$ element.

$${\alpha }_i=\frac{{\sum }_{j=1}^n{b}_{ij}}{{\sum }_{i=1,j=1}^{nn}{b}_{ij}}$$

(8)

where $b_{ij}$ represents the elements in the judgment matrix.

$CI$ (consistency index) and $CR$ (consistency ratio) are calculated with Equations (9) and (10) to verify whether the $\alpha$ is reasonable.

$$CR=\frac{CI}{RI}$$

(9)

where $RI$ is the average random consistency index.

$$CI=\frac{{\lambda }_{max}-n}{n-1}$$

(10)

where ${\lambda }_{max}$ can be obtained by Equation (11) and n is the size of the matrix.

$${\lambda }_{max}=\frac{1}{n}\sum _{i=1}^n\frac{{\left(A\alpha \right)}_i}{{\alpha }_i}$$

(11)

If $CR$ <0.1, it indicates that the eigenvector is reasonable, which can be used as W. With this and Equation (7), the $SI$ of the event node can be obtained [33].

3.3. Scheduling Path

The suppression index of the parent $nod{e}_i$ can be calculated with Equation (12):

$$S{I}_i=\sum _{k=1}^{n}S{I}_k$$

(12)

where ${SI}_k$ denotes the suppression index of child $nod{e}_k$. Due to the different scheduling paths, various combinations of resource scheduling methods can be obtained. From the leaf node to the root node, along the path, the $SI$ of the root node can be calculated. When the $SI$ of the root node is greater than the $SI$ value required to control the fire, it can be considered as an available firefighting resource scheduling path. With these paths, we can know from where and which resources should be allocated.

4. Results

4.1. Scenario

The proposed method was verified with the real field data of Nanjing Laoshan National Forest Park, which is located at 118°30${}^{\prime }$ E, 30°40${}^{\prime }$ N. Generally, the wind speed, temperature, humidity, fuel type, and slope are used to evaluate the scale of the fire and the number of resources required to control the fire. With this, the required suppression index can be obtained. In this paper, we considered a mid-size fire with real historical data. There are three fire stations, M1, M2, and M3, in the area. The fire stations in the study area were obtained from Google Maps, as shown in Figure 4.

The firefighting resources of each fire station are different. Fire stations M1 and M2 have only firefighters and fire engines, and fire station M3 is equipped with drones for firefighting. The distance between the fire station and the fire was calculated based on Dijkstra’s algorithm [35] with DEM data.

Table 5. The number of resources at each fire station and the distance from the fire.

Fire Stations	Fire Engines	Drones	Firemen	Distance from Fire
M1	5	0	15	3 km
M2	8	0	20	10 km
M3	15	3	35	14 km

lists the available fire resources in each fire station.

Most of the fires that occurred in the area were medium- and small-scale. Generally, based on the travel time criterion [29] and firefighting distance criterion [30], there are a few fire stations nearby that should respond to the occurrence of fires and could arrive at the fire field in the required time within a moderate traveling distance. In most cases, the fire could be effectively controlled with the cooperation of a few firefighting stations in this area.

4.2. Result of E-R Tree Model

The goal of the root node is to put out the fire, as shown in Figure 5. The symbols’ meanings are given in

Table 6. Symbols in the E-R tree.

Symbol	Kind	Explanation
Root	Root	Successfully Put Out the Fire
r1	response	Dispatch firefighting resources from M1
r2	response	Dispatch firefighting resources from M2
r3	response	Dispatch firefighting resources from M3
r4	response	Dispatch fire engines with firemen from M1
r5	response	Dispatch firemen from M1
r6	response	Dispatch fire engines with firemen from M2
r7	response	Dispatch firemen from M2
r8	response	Dispatch fire engines with firemen from M3
r9	response	Dispatch firemen from M3
r10	response	Dispatch drones with firemen from M3
e1	event	Fire engines from M1
e2	event	Firemen who operate fire engines from M1
e3	event	Firemen from M1
e4	event	Fire engines from M2
e5	event	Firemen who operate fire engines from M2
e6	event	Firemen from M2
e7	event	Fire engines from M3
e8	event	Firemen who operate fire engines from M3
e9	event	Firemen from M3
e10	event	Firemen who operate drones from M3
e11	event	Drones from M3

. The E-R tree can be divided into three levels. The first level consists of responses by the different fire stations in the early stage of the fire spreading. The second level consists of responses to possible firefighting-resource-dispatching schemes in each fire station. The third level is the event for which the resources should be dispatched with different schemes.

4.3. Scheduling Path of E-R Tree

In order to minimize the number of firefighting resources that should be dispatched, we preferentially selected resources with large $SI$ for scheduling. Figure 6 shows the scheduling paths of the E-R tree, where we can clearly see the resources from stations to the destination.

Table 7. Scheduling paths of different resources.

Scheduling Path	Nodes	Stations	Number of Resources	SI	Meaning
SP 1	e1, e2, r4, r1, root	M1	5	2.43	5 fire engines with 10 firemen from M1
SP 2	e3, r5, r1, root		5	1.98	5 firemen from M1
SP 3	e4, e5, r6, r2, root	M2	8	2.76	8 fire engines with 16 firemen from M2
SP 4	e7, e8, r8, r3, root	M3	6	1.14	6 fire engines with 12 firemen from M3
SP 5	e10, e11, r10, r3, root		3	1.84	3 drones with 3 firemen from M3

summarizes the obtained results of Figure 6. We can know that five scheduling paths are unitized to dispatch firefighting resources to achieve the expected fire suppression effect. The paths are SP1, SP2, SP3, SP4, and SP5 with different colors, respectively.

4.4. Result of MIP

We also used MIP to schedule available fire resources (Section 2.3), as shown in

Table 8. MIP schedules the number of resources.

Resources	Station	Number of Resources
e1	M1	5
e2	M1	10
e3	M1	5
e4	M2	8
e5	M2	16
e6	M2	0
e7	M3	6
e8	M3	12
e9	M3	0
e10	M3	3
e11	M3	3

. The results achieved by the E-R tree-based resource scheduling model were the same as the MIP model. Compared with MIP, the relationship between various resource-dispatching processes was clear and intuitive for firefighters to make decisions.

From the experimental results, it can be seen that the M1 fire station plays a larger role than the M2 and M3 fire stations, and the fire engines’ suppression of fire from the fire station is relatively obvious compared to other firefighting methods. In the actual firefighting activities, the number can also be increased appropriately to improve the effectiveness of fire suppression and decrease the impact that wildfires have.

5. Conclusions

Dispatching firefighting resources is a critical issue as far as wildfire fighting is concerned. Thus, it is necessary to improve the ability of decision-makers to dispatch firefighting resources in a more efficient way so that the fire can be controlled in time. In this paper, a new E-R tree-based resource scheduling model to dispatch firefighting resources based on the fire suppression index (SI) was proposed. The proposed model not only achieves available resource scheduling, but also clearly shows the relationship between various resource-dispatching processes. In order to validate the proposed method, we compared it with the widely used MIP model by using actual data. The results showed that the E-R tree-based resource scheduling can effectively schedule resources as well as the MIP model. Moreover, the relationship between various resources dispatched in the proposed model was clearly displayed. The decision-makers can clearly see the resources scheduled from each station to the fire field. This is helpful for the decision-makers to schedule the resources quickly.

Although our work here was presented in the context of firefighting resource scheduling, the model may be deployed in other resource-scheduling scenarios as well. The root node of the E-R tree denotes the goal of resource scheduling. For example, for the dispatching of emergency supplies, its goal can be to minimize the loss from the emergency [36]. According to variable resources at each site, the E-R tree model can be constructed, and the decision-maker can dispatch the resources easily and intuitively. As such, the proposed method could have a wider scope of applicability.

Although the E-R tree scheduling model can schedule the resources in a simple and intuitive way, since the proposed method is partly based on AHP, subjectivity may exist. Currently, there are other methods, such as fuzzy AHP [37] and multi-layer AHP [38], that can be used to assign weights to the factors in a more accurate way. We intend to investigate this aspect in our future work.