In this paper we propose a stochastic integer programming optimization model to determine the optimal location and number of rain water collectors (RWCs) for forest firefighting. The objective is to minimize expected total cost to control forest fires. The model is tested using a real case and several additional realistic scenarios. The impact on the solution of varying the limit on the number of RWCs, the RWC water capacity, the aircraft capacity, the water demands, and the aircraft operating cost is explored. Some observations are that the objective value improves with larger RWCs and with the use of aircraft with greater capacity.
Currently, many forested areas are being destroyed due to an inefficient system for controlling fires. The problem is due to the absence of nearby water resources, the low availability of aircraft to transport water, and poor logistics. Additionally, because of the growth of urban municipalities, fires occur near populated municipalities. According to the Earth Policy Institute [
The magnitude, speed, and difficulty to control fires cause loss of natural and economic resources. In Mexico, around 7,000 forest fires burned more than 340,000 hectares in 2012 and 413,216 hectares in 2013, according to the report of the National Forestry Commission of Mexico [
The deployment of firefighters, pumper trucks, light vehicles, open firewall machines, and gondolas to transport bulldozers all contribute to the cost of extinguishing a forest fire. The list of human and material resources that can be mobilized to a forest fire is long and varied. A light vehicle used to combat fire costs approximately 288 euros an hour as discussed by Cáceres [
Before a facility can be constructed, quality sites must be identified and proper facility capacity specifications must be determined. Facilities are expected to operate for prolonged times; therefore, defining locations for new facilities is an important strategic challenge as discussed by Weber and Friedrich [
Since the early 1960s facility location problems are often treated as minimization models for logistics costs. For instance, Hakimi [
On the other hand, research in the area of response-time minimization has gained importance in a market characterized by demanding customers and rigid competition among firms. Organizations often explore opportunities for customer satisfaction through response time minimization as a competitive strategy, which is evident in initiatives such as the lean philosophy as exposed by Womack and Jones [
As mentioned above, several different objective functions have been formulated to make location theory models tractable in numerous applications. Unfortunately, most problems are classified as NP hard; thus, the resulting models are exceptionally difficult to solve optimality. Moreover, most problems require integer programming formulations. Due to the complex formulations, computational requirements are elevated.
As noted by Averbakh and Berman [
Several types of water sources, such as hydrants in urban areas and lakes in rural areas, are used to control fires. Even nonconventional sources, such as swimming pools, are used in rare cases. In addition to these water resources, Garza-Ramirez [
RWCs store rainwater for a specific use before it reaches the underground aquifers. Uses include water for gardens, livestock, irrigation, and even for human consumption. In this paper, the focus is on RWCs, which can be built to satisfy human water demand and also to serve as water sources for supplying firefighting helicopters. According to the report of the United States Environmental Protection Agency (EPA) [
Ground-level rain water collector (RWC) design. Source: Water and Life Program, Tecnológico de Monterrey, Mexico.
Giroz and Julio [
Different approaches to improve wildfire containment have been attempted. Many models that address aircraft routing for firefighting have been developed. In 1980, Paredes [
In this paper, we present a stochastic integer programming model to find the optimal number of RWCs to be constructed and their locations in order to minimize the expected cost to construct RWCs and to control fires. The costs considered are construction cost for RWCs, cost for ecosystem damage, and logistic costs. The model was applied to the state of Nuevo Leon, Mexico. Municipalities (customers) require water sources (facilities) to meet demand (quantity of water to control the fires). To date, no stochastic models found in the literature consider cost minimization for the use of rain water collectors for firefighting.
The remainder of the paper is organized as follows: Section
This section presents a stochastic optimization model to minimize expected cost derived from RWC constructions, ecosystem damages, and firefighting logistics such as helicopter flight costs. In this model, the objective is to minimize the expected total cost. The model utilizes discrete decision variables. Before proceeding, the assumptions for the model are as follows:
The notations are defined in notations section.
The problem is as follows:
The first term of the objective function (
Our model and solution method has advantages over the current decision making method. Currently, the decision to place an RWC is based only on a municipality’s demand for water. The suitability of the location for firefighting purposes is not currently taken into account. RWCs located using this approach can be used for firefighting but might not be in an advantageous location for this purpose. In contrast, our model explicitly considers the suitability of the RWC location for firefighting. To take into account the municipalities’ needs for water, the candidate locations can be restricted to locations that satisfy these requirements.
Our model also has advantages over a deterministic formulation. A deterministic formulation can be obtained by deleting the
In this paper, we use the approximation for forest loss cost per hectare proposed by Armando [
To calculate the distance from each potential RWC construction site to the fire location, we consider the potential site coordinates,
In order to compute the total distance traveled by an aircraft,
This computation assumes that all water in the RWC is used. Although it is possible for demand to be satisfied in fewer trips, in practice it is more likely to require the full water capacity. In this way the model is conservative and plans for a maximal number of trips.
The probability of a fire occurring,
To validate the model, the state of Nuevo Leon, Mexico, was used as a case study. According to the National Forestry Commission, during 2011, Nuevo Leon suffered thirty-five fires which affected 707 hectares of forest [
Probabilities of fire per municipality.
Municipality | Number of fires |
|
---|---|---|
Galeana | 77 | 0.430 |
Santiago | 32 | 0.179 |
Monterrey | 21 | 0.117 |
Santa Catarina | 16 | 0.089 |
Villaldama | 9 | 0.050 |
Salinas Victoria | 6 | 0.034 |
Bustamante | 5 | 0.028 |
San Pedro Garza García | 4 | 0.022 |
García | 3 | 0.017 |
San Nicolás de los Garza | 1 | 0.006 |
General Escobedo | 1 | 0.006 |
Carmen | 1 | 0.006 |
Abasolo | 1 | 0.006 |
Hidalgo | 1 | 0.006 |
China | 1 | 0.006 |
|
||
Total | 179 | 1.000 |
Let
Water demand to extinguished fires can be obtained using Royer and Floyd [
Water demand for each municipality.
Municipality | Demand (m3) |
---|---|
Santiago | 88,307.8 |
Galeana | 62,700 |
Villaldama | 29,090 |
Salinas Victoria | 22,640 |
Santa Catarina | 3,442 |
China | 2,700 |
Garcia | 620 |
San Pedro Garza Garcia | 412 |
Monterrey | 401.6 |
Bustamante | 232.02 |
San Nicolas de los Garza | 40 |
Abasolo | 40 |
Carmen | 20 |
Hidalgo | 20 |
General Escobedo | 0.01 |
|
|
Total | 210,665.43 |
An important point is that this region of Nuevo Leon has nine existing water sources. These water sources are not RWCs and have different capacities. Real data for these sources is utilized as an input to our model. Potential sites for RWCs are generated at random within the study region. The region was delimited by the convex hull given by the outer municipalities.
Using Velasco Molina [
Based on historical data and using (
Forest loss cost per municipality.
Municipality | FLC (US Dollars) |
---|---|
Santiago | 11,811.31 |
Santa Catarina | 920.74 |
San Pedro Garza Garcia | 440.85 |
Monterrey | 81.85 |
Galeana | 3,485.18 |
San Nicolas de los Garza | 171.20 |
General Escobedo | 0.04 |
Carmen | 85.60 |
Garcia | 884.54 |
Abasolo | 171.20 |
Hidalgo | 85.60 |
Salinas Victoria | 16,150.06 |
Villaldama | 13,834.08 |
Bustamante | 198.61 |
China | 11,556.14 |
The cost per unit distance,
In order to gain insight into the behavior of the model, a numerical study was performed. The study consisted of solving the model with varying values of
Varying the maximum number of RWCs for the case study.
The number of water sources to be constructed in the optimal solution is 29. A total deficit of 12,665.43
An analysis of the selected RWCs was performed. This is relevant information to consider because it reflects the possible layout for the construction and utilization of RWCs. Table
Construction and selection of RWCs.
RWC | USE |
---|---|
1 | 1 |
2 | 1 |
3 | 1 |
4 | 1 |
5 | 1 |
6 | 1 |
7 | 1 |
8 | 1 |
9 | 1 |
10 | 1 |
11 | 1 |
12 | 1 |
13 | 1 |
14 | 1 |
15 | 1 |
16 | 1 |
17 | 0 |
18 | 0 |
19 | 0 |
20 | 0 |
21 | 0 |
22 | 0 |
23 | 0 |
24 | 0 |
25 | 1 |
26 | 1 |
27 | 1 |
28 | 0 |
29 | 0 |
30 | 1 |
31 | 1 |
32 | 0 |
33 | 1 |
34 | 0 |
35 | 0 |
36 | 1 |
37 | 1 |
38 | 0 |
39 | 0 |
40 | 0 |
41 | 1 |
42 | 1 |
43 | 1 |
44 | 1 |
45 | 0 |
46 | 0 |
47 | 0 |
48 | 1 |
49 | 0 |
50 | 0 |
|
|
Total | 29 |
Based on the RWCs selected in the optimal solution, the optimal layout for the construction of the RWCs is shown in Figure
Optimal solution for the case study.
As can be observed, the construction of RWCs is concentrated near Santiago and Galeana, which are the municipalities with greater demand. Table
RWC assignments.
Municipality | RWCs assigned | Deficit (m3) |
---|---|---|
Abasolo | 0 | 40 |
Bustamante | 0 | 232.02 |
Carmen | 0 | 20 |
China | 0 | 2700 |
Galeana | 1, 2, 3, 6, 7, 9, 10, 12, 13, 14, 25, 27, 30, 36, 37, 41, 42, 43, 48 | 2700 |
Garcia | 0 | 620 |
General Escobedo | 0 | 0.01 |
Hidalgo | 0 | 20 |
Monterrey | 0 | 401.6 |
Salinas Victoria | 4, 15, 16, 26, 31, 33, 44 | 1640 |
San Nicolás de los Garza | 0 | 40 |
San Pedro Garza García | 0 | 412 |
Santa Catarina | 11 | 442 |
Santiago | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 25, 26, 27, 30, 31, 33, 36, 37, 41, 42, 43, 44, 48 | 1307.8 |
Villaldama | 4, 5, 11, 15, 16, 26, 31, 33, 44 | 2090 |
Since FLC is an estimated value, it is important to understand the impact of its variation. For this, we lowered and increased FLC by 50% and 25%, respectively, with respect to the original data. All other values were kept unchanged. Figure
Experiments outcome.
FLC | Optimum L | Objective function |
---|---|---|
−50% | 29 | 11,552,844 |
−25% | 29 | 14,722,360 |
Original | 29 | 17,891,875 |
+25% | 29 | 21,061,390 |
+50% | 29 | 24,230,905 |
Varying FLC for the case study.
An observation from the experiments is that no deficit exceeds the capacity of an RWC. This means that the construction cost plus the logistic costs involved in the assignment of a new RWC is higher than the forest loss cost associated with the water deficit. Another observation is that most of the RWCs are constructed near the municipalities with higher water demands. This is because greater demand increases the number of required round trips, which consequently increases logistics cost. Therefore, reducing the distance to higher demand municipalities reduces logistics costs. Another observation is that the objective value increases as FLC is increased, but the number of RWCs to be constructed does not increase. This means that, in this case study, the construction and logistics costs (keeping in mind that logistics costs depend on the location) for an additional RWC are higher than the increase in forest loss cost, even when FLC is increased by 50%.
In order to explore alternative scenarios, an additional set of instances was solved. These instances were based on the real case study but with changes to input values such as the RWC capacities, the aircraft capacities, and the water demands. Scenario 0 represents the real data used in the case study presented above. For scenarios 1 and 2, we lowered and increased demand to control fire by 20%, respectively. For scenarios 3 and 4, the RWC capacity was decreased and increased by 1,000 liters, respectively. For scenarios 5 and 6, the helicopter capacity was increased to 3,000 liters and lowered by 500 liters. Scenarios 7 and 8 vary the cost per unit distance by ±20%. To solve the proposed model, we used OPL to call Cplex 12.0. Table
Numerical data used in each scenario.
Scenario | Number of cities | Capacity of RWC | Capacity of helicopters | Demand to control fire ( |
Cost per unit distance ( |
---|---|---|---|---|---|
1 | 15 | 3,000 liters | 1,500 liters | −20% | 0% |
2 | 15 | 3,000 liters | 1,500 liters | +20% | 0% |
3 | 15 | 2,000 liters | 1,500 liters | 0% | 0% |
4 | 15 | 4,000 liters | 1,500 liters | 0% | 0% |
5 | 15 | 3,000 liters | 3,000 liters | 0% | 0% |
6 | 15 | 3,000 liters | 1,000 liters | 0% | 0% |
7 | 15 | 3,000 liters | 1,500 liters | 0% | −20% |
8 | 15 | 3,000 liters | 1,500 liters | 0% | +20% |
Since Nuevo Leon has 9 existing water sources, it is set as lower bound to optimize strategic planning. The maximum number of RWCs allowed,
Testing results.
Scenario | Maximum RWC allowed | Number of RWCs selected | Objective function | Deficit per year | Feasibility |
---|---|---|---|---|---|
0 | 9 | 9 | 188,864,744 | 105,665 | YES |
20 | 20 | 73,649,199 | 39,665 | YES | |
30 | 29 | 17,891,875 | 12,665 | YES | |
|
|||||
1 | 9 | 9 | 131,984,967 | 75,532 | YES |
20 | 20 | 33,624,708 | 21,532 | YES | |
30 | 29 | 15,029,308 | 12,532 | YES | |
|
|||||
2 | 9 | 9 | 248,165,989 | 138,798 | YES |
20 | 20 | 128,861,206 | 66,798 | YES | |
30 | 29 | 49,648,355 | 21,798 | YES | |
|
|||||
3 | 9 | 9 | 229,258,656 | 134,665 | YES |
20 | 20 | 144,908,486 | 76,665 | YES | |
30 | 30 | 83,645,882 | 38,665 | YES | |
50 | 44 | 77,669,925 | 6,665 | YES | |
|
|||||
4 | 9 | 9 | 156,246,190 | 90,665 | YES |
20 | 20 | 29,897,080 | 22,665 | YES | |
30 | 22 | 13,273,840 | 14,665 | YES | |
|
|||||
5 | 9 | 9 | 187,425,499 | 105,665 | YES |
20 | 20 | 70,350,783 | 39,665 | YES | |
30 | 29 | 14,374,364 | 12,665 | YES | |
|
|||||
6 | 9 | 9 | 190,303,988 | 105,665 | YES |
20 | 20 | 76,947,615 | 39,665 | YES | |
30 | 29 | 21,409,385 | 12,665 | YES | |
|
|||||
7 | 9 | 9 | 188,432,970 | 105,665 | YES |
20 | 20 | 72,659,674 | 39,665 | YES | |
30 | 29 | 16,836,621 | 12,665 | YES | |
|
|||||
8 | 9 | 9 | 189,296,517 | 105,665 | YES |
20 | 20 | 74,638,723 | 39,665 | YES | |
30 | 29 | 18,947,127 | 12,665 | YES |
The results from these instances yield some insights. One, which is to be expected, is that when there is a deficit of water, forest loss cost is incurred which yields higher objective values. This occurs because the FLC is much more than the construction cost of an RWC. A second insight is that an improved objective function is found for RWCs of greater capacity, and this is due to an increase of water availability from the RWCs that are selected. Thus, more water is available from the optimal locations, as opposed to having to construct RWCs in suboptimal locations in order to meet demand. Consequently, fewer RWCs need to be constructed. Another observation is that decreasing demand improves the objective function because logistic costs are reduced since fewer trips are needed to satisfy demand. It can also be observed that as the helicopter capacity increases, the objective function is improved due to the reduced number of required trips. Additionally, higher costs per unit distance yield higher objective values. This occurs because the logistic cost increases. A second insight is that a very high cost per unit distance inflicts logistic costs in such a manner that it is preferable to incur FLC. In other words, it is better, from a financial perspective, to let the forest burn.
In this paper we propose an optimization model with the objective of minimizing expected total cost to control forest fires. A model is developed based on stochastic integer programming and solved using Cplex and OPL. The model is tested using a real case and several additional realistic scenarios. The impact of varying the limit on the number of RWCs, the RWC capacities, the aircraft capacities, and the water demands on the solution quality is explored. Some observations are that the objective value improves with larger RWCs and with the use of aircraft with greater capacity.
Some recommendations for further research are to extend the model to consider multiple simultaneous fires. It is also possible that, due to location constraints, an RWC selected by the model cannot be placed in that particular position site. Therefore, models that consider ground topography such as mountains, streets, houses, buildings, and biodiversity, and property lines, among others, could be proposed. Updating of costs for forest lost and RWC is recommended in order to build more realistic cost-benefit scenarios. As well, models that consider social benefits associated to stored water in RWCs for local people in forest areas are necessary.
Number of candidate RWC locations
Number of municipalities
Number of candidate locations for RWCs
Limit on number of RWCs that can be constructed
Probability of having a fire in municipality
Total travel distance between municipality
Capacity of water source
Construction cost of RWC
Forest loss cost per
Cost to mobilize a helicopter per unit distance
Water demand in
Unsatisfied water demand in
The authors declare that there is no conflict of interests regarding the publication of this paper.