A Hybrid algorithm applied to facility location for forest fire fighting considering budget constraints M. A. Diaz-Romero V. Alarcon-Aquino Department of Computing, Electronics and Mechatronics Universidad de las Americas Puebla Sta. Catarina Martir, Cholula, Puebla. 72810 MEXICO {minerva.diazro,vicente.alarcon}@udlap.mx

Abstract—This paper develops an integrated system for forest fire fighting facility location. We propose a mathematical model to deploy available fire fighting resources in proper positions so that any forest fire can be attacked within a specified response time. The proposed model considers budget constraints. This location problem is a variant of the well-known Maximal Covering Location Problem (MCLP) which is known to be NP-hard. We propose a hybrid algorithm that combines GRASP and Tabu Search procedures. According to preliminary computational experiments the hybrid algorithm provides good quality solutions with a reasonable amount of computer effort. This solution can give support to the decision maker for determining the location of fire-fighting resources.

I. I NTRODUCTION Across the globe, wildfire-related destruction appears to be worsening despite increased fire suppression expenditure. At the same time, wildfire management is becoming increasingly complicated owing to factors such as an expanding wildland-urban interface, interagency resource sharing and the recognition of the beneficial effects of fire on ecosystems. Operations research refers to the use of analytical techniques such as mathematical modelling to analyse interactions between people, resources and the environment to aid decisionmaking in complex systems [1]. We consider a variant of the maximum covering location problem which is a combinatorial optimization problem. The maximal covering location problem has been applied to numerous facility location problems, including rural health centers [2], emergency vehicles [3], and commercial bank branches [4], as well as other applications [5]. Optimization models and methods have been applied to the solution of forest planning problems for over 30 years, and during this time the nature of the problems have evolved. The diversity of decision problems and the size of planning tasks has greatly increased [6]. There are several studies related to forest fires, an approach for detecting smoke columns from outdoor forest video sequences in [7] is used to reduce the number of false alarms. Experimental results show that multi-resolution wavelet analysis is more accurate than the traditional low-pass filters on this application. There is another work, where a mathematical model is presented for predicting the maximum potential

J. A. Diaz-Garcia Department of Actuarial Science, Physics and Mathematics Universidad de las Americas Puebla Sta. Catarina Martir, Cholula, Puebla. 72810 MEXICO [email protected]

spot fire distance from an active crown fire [8]. Absence of management has in turn been blamed for increasing the risk of fire, thus forming a vicious circle of low profitability, little management and high risk of fire. Developing forest planning tools that maximize both economic objectives and fire resistance could help to revive the forest sector in the region and generate long-term fire prevention strategies [9]. Forest fires have received increased public attention worldwide in the time span of the last few years, due to their significant short and long term threats to forest ecosystems and to public safety and property. Six million square kilometers of forest land have been lost around the world, in less than 200 years, mainly due to forest fires. One of the mechanisms to confront the problem is an effective strategy for forest fire management [10]. Wildfires management has been used in the development of several analysis methodologies based on operation research techniques that serve as tools for policy creation. This is reflected on the growing number of applications, some of which are discussed in [11]. The goal of controlling wildfires is to minimize the negative impact they cause. This can be achieved through early detection and fire attack soon after it is informed or through an effective control of fires [12]. There is another work where a specific mix of fire-fighting is identifited in order to minimize the Cost Plus Net Value Change (C+NVC) [13]. Different countries have addressed the issue of fire management such as Australia [14] and Grecia [15]. Dimopoulou y Giannikos present a system that consists of a geographical information system (GIS) module, a mathematical programming (MP) module and a simulation module. This work is an extension of [15], where only one fire fighting resource is considered. There exists different kinds of fire fighting resources such as brigades, Water Pump, chainsaw, helicopter, fire truck. In this work we consider three different types of resources: brigades, fire truck, helicopter and a budget constrain. Therefore we determine not only the location of each resource but also the fire fighting resource that should be positioned in each location. To the best of our knowledge, this version of the problem has not been studied in the literature. In [15] only a give number of only one type of fire fighting

resource is considered. II. P ROBLEM DEFINITION The main goal of this work is to propose efficient solution algorithms for the variant of the Maximal Covering Location Problem (MCLP) with multiple resources and budged constraints considered to locate fire fighting resources. The objective of the proposed formulation of the problem is to maximize the weighted sum of demand points covered in a protected area with different resource types. Considering the number of available resources, deployment is made considering a budget constraint and the arrival time limit to extinguish the fire (which varies for each zone). Namely, the task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. Let A denote the index set of demand points covered by one or more resources. That is, those demand points that are within the coverage radius of those resources that have been previously located. Let also Sk denote the index set of selected resource locations or type k ∈ K. We propose a mathematical programming model that determines the resource deployment in such a way that any fire can be attacked within a certain time limit. The deployed resources must cover the largest geographical area that was previously classified for its level of importance, the MCLP is known to be NP-Hard. Currently, all known algorithms for NP-Hard problems require time that is superpolynomial in the input size, and it is unknown whether there are any faster algorithms. There are several heuristic methodologies that can be used to find near optimal solutions with a reasonable ammount of computational effort for difficult optimization problems. A heuristic is an algorithm that works reasonably well in many cases, but for which there is no proof that it is always both fast and effective. Metaheuristic approaches are often used since they provide mechanisms to avoid local optimality. In this work, we describe a greedy randomized adaptive search procedure (GRASP) combined with Tabu Search for the MCLP with budget constraint that finds approximate good, though not necessarily optimal solutions, for location of fire-fighting resources. Approximate algorithms cannot guarantee optimality of the solutions they return, but empirically they have often been shown to return good solutions with a reasonably ammount of computational effort. III. M ATHEMATICAL MODEL The MCLP has a compact integer programming formulation, first described in [16]. Covering models for emergency service deployment typically assume that model parameters are known with certainty, such as number of candidate locations, the number of demand points and the cost per unit of resources. The model considered in this work differs from classical MCLP in that it considers multiple resources, different time limits for each of them and budget constraints (i.e. the number of facilities to locate is not fixed beforehand). The following formulation

characterizes the objective function and constraints that are required to identify the location of fire-fighting resources that maximize the weighted sum of the covered demand points. Sets of indices: J: Set of demand points. K: Set of resources. Ik : Set of candidate locations of resources of type k. Ck : Cost per unit of resource of type k. Parameters: N (j, k): Set of candidate locations that can cover demand point j by resources of type k ; j ∈ J, k ∈ K wj : weight according the need for protection of the demand points j; j ∈ J P : Budget Decision variables:

xik =

yj =

  1  

if a resource of type k is located in location i;i ∈ Ik , k ∈ K , otherwise.

 

if demand point j is covered by at least one resource, j ∈ J, otherwise.

0   1 0

The model is formulated as follows: X Maximize wj yj

(1)

j∈J

subject to: yj −

X

X

xik ≤ 0 ∀j ∈ J

k∈K i∈N (j,k)

XX

(2)

Ck xik ≤ P

k∈K i∈Ik

xik , yj ∈ {0, 1} ∀k ∈ K, ∀j ∈ J, ∀i ∈ Ik

(3)

(4)

N (j, k) is the set of candidate locations that can cover demand point j by a resource of type k. More specifically, N (j, k) = {i|t(i, j, k) ≤ T M ax(j)}

(5)

where T M ax(j) is the time limit for demand point j as calculated depending on the class of demand point j. Finally, t(i, j, k) is the time required for a resource of type k to travel from candidate location i to demand point j. The objective function (1) represents the weighted sum of demand points that are covered. Different weights of wj express the different requirements for protection of the demand points. Since some

of the demand points may not be covered at all due to the limited number of resources, the demand points that require greater protection are those who will have greater weight. Constraints (2) specify that a demand point j is covered only if at least one resource is located in a point that can cover j within standard response time. The constraint (3) ensures that budget is not exceeding. Usually, for many mixed integer programming problems, there is no known polynomial-time algorithm and there are little chances that one will ever be found [17]. Some of these use the classical Branch and Bound and more sophisticated concepts such as Branch and Price or Branch and Cut to reach the optimal solution, these algorithms are known as exact methods. The Maximum Covering Location Problem is known to be NP-Hard. Therefore, we describe the Hybrid algorithm for the proposed variant of the MCLP to locate fire-fighting resources that can be used to obtain near-optimal solutions. IV. P ROPOSAL OF GRASP AND TABU S EARCH FOR MCLP WITH BUDGET CONSTRAINT (H YBRID ALGORITHM ) A. General description of GRASP GRASP [18] is a meta-heuristic that has been applied to a wide range of combinatorial optimization problems, including set covering [19]. GRASP is an iterative procedure, with a feasible solution obtained at each independent GRASP iteration [20]. Each GRASP iteration consists of two phases, a construction phase and a local search phase. The best overall solution is kept as the result. GRASP typically consists of iterations made up from successive constructions of a greedy randomized solution and subsequent iterative improvements of it through a local search. The greedy randomized solutions are generated by adding elements to the problem’s solution set from a list of elements ranked by a greedy function according to the quality of the solution they will achieve. To obtain variability in the candidate set of greedy solutions, well-ranked candidate elements are often placed in a Restricted Candidate List (RCL), and chosen at random when building up the solution. This kind of greedy randomized construction method is also known as a semigreedy heuristic [20]. In the construction phase, a feasible solution is iteratively constructed, one element at a time. At each construction iteration, the choice of the next element to be added is determined by ordering all elements in a candidate list with respect to a greedy function [19]. Since the solutions generated by the constructive phase of the GRASP procedure are not guaranteed to be locally optimal with respect to simple neighborhood definitions, it is usually beneficial to apply a local search to attempt to improve each constructed solution. 1) Construction phase: In this section, we describe in detail the construction phase of the proposed GRASP algorithm. We provide a candidate definition (for the restricted candidate list), an adaptive greedy function, and specify the candidate restriction mechanism (see Algorithm 1). Let A denote the index set of demand points covered by one or more resources. That is, those demand points that

are within the coverage radius of those resources that have been previously located. Let also Sk denote the index set of selected resource locations or type k ∈ K. The GRASP construction phase is a greedy algorithm. It is an iterative approach. At each iteration a potential site i? ∈ Ik \ Sk is selected to locate a resource of type k ? , k ? ∈ K, and the set of demand points within the coverage radius of the resource ? k ? located at i? are updated, that is, A ← A ∪ Jik? . Where Jik = {j : t(i, j, k) ≤ T M ax(j)}. The selected resource k ? and its location i? are randomly selected from a restricted candidate list, RCL, that contains the best candidates with respect to the following greedy function, P

wj

j∈Jik \(Jik ∩A)

∀k ∈ K, i ∈ Ik \ Sk (6) Ck where, gik value measures the weigth per unit cost covered by a resource of type k located at site i. The restricted candidate list RCL contains pairs of the form (i, k), k ∈ K, i ∈ Ik \ Sk whose greedy value is greater than or equal to a threshold value, gik =

T ← gmax − α(gmax − gmin )

(7)

where gmax = maxk∈K,i∈Ik \Sk {gik }, gmin = mink∈K,i∈Ik \Sk {gik } and the α ∈ (0, 1) parameter controls the greediness or randomness of the greedy procedure. The greedy procedure terminates when the available budget is less than mink∈K {Ck }. Algorithm 1 Construction phase Require: K Set of resources. Ik Set of candidate locations of resources of type k Ck Cost per unit of resource of type k. P Budget α Randomness algorithm. Ensure: A Set of demand points covered. Sk Resources location selected of type k z: Total weight covered demand points (f.o) 1: Sk ← ∅ 2: A ← ∅ 3: while P > mink∈K {Ck } do P wj

j∈J k \(J k ∩A)

4: 5: 6: 7: 8: 9: 10: 11:

i i ∀k ∈ K, i ∈ Ik \ Sk gik = Ck T ← gmax − α ∗ (gmax − gmin ) RCL ← {(i, k) : gik ≥ T } Randomly selected a pair (i∗, k∗) of RCL Sk ∗ ← Sk ∗ ∪{i∗} A ← A ∪ Jik P ← P − Ck∗ end while

2) Local search phase: Local search are effective improvement algorithms. They start from some given solution and try to find a better solution in an appropiately defined neighborhood of the current one (see Algorithm 2). A local search

procedure is an iterative approach. At each iteration the current solution is replaced by the best neighbour solution, when the objective function value of the best neighbour solution is better than the objective function value of the best solution found so far. The iterative procedure terminates when the objective function value of all neighbors solutions is worst that the objective function value of the current solution. Local search algorithms usually terminates with a local optimum. The choice of an appropriate neighborhood structure is crucial for the performance of a local search algorithm and often has to be done in a problem specific way. Given a feasible solution s = {S1 , S2 , . . . , Sk } we use the following neighborhood structure: N (s) = {s0 = {S10 , S20 , . . . , Sk0 } : ∃!k ? ∈ K, k1 ∈ Sk? , k2 ∈ Ik? \ Sk? , Sk0 ? = (Sk? ∪ {k2 }) \ {k1 }} N (s) is the set of solutions that can be reached from the current one by changing the location of one resource from the selected site for that resource to a non-selected site for that resource type. The pseudo-code for the local search phase is presented in Algorithm 2. Finally, the pseudo-code for the GRASP is shown in Algorithm 3. Algorithm 2 Local search phase Require: K Set of resources. Ik Set of candidate locations of resources of type k Ck Cost per unit of resource of type k. A Set of demand points covered. Sk Resources location selected of type k z: Total weight covered demand points (f.o) Ensure: z∗ : Best-improvement 1: z∗ ← 0 2: for all k ∈ K do 3: for all j ∈ Sk do 4: for all t ∈ Ik \ Sk do 5: Sk ← Sk − {j} 6: Sk ← Sk + {t} P 7: a ← argmaxi∈Ik ( wj )

of the construction phase is to produce a good initial solution to be improved by the local search procedure. It is empirically known that simple local search techniques perform better if they start with a good initial solution. Algorithm 3 GRASP algorithm Require: Let z∗ be the incumbent, α ∈ (0, 1) and γ set of selected resources Ensure: 1: repeat 2: γ ← ConstructionP hase(α) 3: γ ← LocalSearch(γ) 4: if z(α) > z∗ then 5: z∗ ← z(γ) 6: γ∗ ← γ 7: end if 8: until Stop Criteria

B. Tabu Search The Tabu Search (TS) is a metaheuristic approach whose distinguishing feature is its use of adaptive memory and special associated problem-solving strategies. The philosophy of tabu search is to derive and exploit a collection of intelligent problem solving strategies, based on implicit and explicit learning procedures to exploit an adaptive memory framework. As defined in the GRASP algorithm, let also Sk denote the index set of selected resource locations or type k ∈ K. At each iteration a potential site i? ∈ Ik \ Sk is selected to locate a resource of type k ? , k ? ∈ K, and the set of demand points within the coverage radius of the resource k ? located ? at i? are updated, that is, A ← A ∪ Jik? (see Algorithm 4). The selected resource k ? and its location i? are selected from a list (selecting the largest value), that contains the best candidates, that does not contain tabu attributes, with respect to the function described in (6). The list of tabu attributes is defined by tuples (i, k), where i, k is selected to locate a resource of type. The tenure of a move in the tabu list is of t=5.

j∈Jik \(Jik ∩A)

8: 9: 10: 11: 12: 13: 14: 15:

z1 ← z + a if z1 > z∗ then BestRecur ← t z∗ ← z1 end if end for end for end for

The GRASP construction phase described in Section IV.A computes a feasible set of facility locations that is not necessarily locally optimal with respect to the neighborhood structure. Consequently, local search can be applied with the objective of finding a locally optimal solution that may be better than the constructed solution. In fact, the main purpose

C. Hybrid algorithm In this procedure, we find an initial solution with the GRASP algorithm. The tabu search is used to improve the solution obtained by the GRASP proposed in this work. Tabu search can be used for solving combinatorial optimization problems. First, the tabu search starts with an initial solution, in our case, the initial solution is obtained by the algorithm GRASP. The pseudo-code for the hybrid algorithm is similar to the pseudo-code for the GRASP. The difference is that instead of applying the local search we use tabu search (γ ← LocalSearch(γ)), is replaced by γ ← T abuSearch(γ). The pseudo-code for the Hybrid Algorithm is shown in Algorithm 5.

Algorithm 4 Tabu Search Require: K Set of resources. Ik Set of candidate locations of resources of type k Ck Cost per unit of resource of type k. P Budget Cost Total cost Ensure: A Set of demand points covered. Sk Resources location selected of type k z: Total weight covered demand points (f.o) 1: Sk ← ∅ 2: A ← ∅ 3: repeat 4: Selected ← false 5: while not Selected do P wj

6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:

j∈J k \(J k ∩A) i i

∀k ∈ K, i ∈ Ik \ Sk gik = Ck Selected a pair (i∗, k∗) of max(gik ) if (i∗, j∗) does not contain tabu attributes then Selected ← true Sk ∗ ← Sk ∗ ∪{i∗} A ← A ∪ Jik Cost ← Cost + Ck ∗ Update Tabu List end if end while while Cost ≤ P do Selected ← false while not Selected P do ai,k ← wj j∈Jik \(Jik ∩A)

20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30:

Selected a pair (i∗, k∗) of min(ai,k ) if (i∗, k∗) does not contain tabu attributes then Selected ← true Sk ∗ ← Sk ∗ \{i∗} A ← A \ Jik Cost ← Cost − Ck ∗ Update Tabu List end if end while end while until StopCriteria

Algorithm 5 Hybrid algorithm Require: Let z∗ be the incumbent, α ∈ (0, 1) and γ set of selected resources Ensure: 1: γ ← GRASP (α, γ) 2: repeat 3: γ ← T abuSearch(γ) 4: if z(α) > z∗ then 5: z∗ ← z(γ) 6: γ∗ ← γ 7: end if 8: until Stop Criteria

V. E XPERIMENTAL RESULTS A. Randomly generated maximum covering problems The problem instances generator is based in [20]. We first describe the problem generator. The size of the problem is determined by • m: the number of demand points. • n : the number of potential resources locations. In this work we consider three types of fire-fighting resourses: 1) brigade, 2) fire truck, and 3) helicopter. For each resource, we determine the coverage radius. Let rmax be the maximum distance between a resource location and any demand point covered by the resource location. • rmaxBrigade: distance between a resource of type 1 (brigade) location and any demand point. • rmaxF ireT rucks: distance between a resource of type 2 (FireTrucks) location and any demand point. • rmaxHelicopter: distance between a resource of type 3 (helicopter) location and any demand point. Demand point weights are distributed uniformly in the interval [rmin,rmax], where rmax = 1 and rmin = 0, the demand points are located uniformly in the unit square. n of the m demand points are selected at random to be potential resources locations. A potential resource location located at (xr , yr ) covers a demand point located at (xd , yd ) if p they are within a distance r of each other, i.e., if (xr − xd )2 + (yr − yd )2 ≤ rmax. The experiment was done on problems of dimension m = 10, 000 and n = 2000. Five instances were generated for each problem combination if n, m, and rmax. Demand point weights were generated at random uniformly in the interval [1, 10000]. The indexes for rmaxBrigade, rmaxF ireT rucks and rmaxHelicopter are proposed according to the coverage radius of each resource (0.0040, 0.0056, 0.0200). To allow for different levels of coverage, three levels of rmax were used: • rmaxBrigade : rmaxb := (rmax − rmin) × 0.0040 • rmaxF ireT rucks: rmaxm := (rmax−rmin)×0.0056 • rmaxHelicopter: rmaxa := (rmax − rmin) × 0.0200 B. Computational results The solutions generated by the proposed Hybrid Algorithm are shown in Table I. For each problem, the table reports its name (column 1, Data set), the number of demand points that can be covered by at least one potential resource site, J (column 2) denotes the set of demand points, Ik (column 3) is the set of candidate locations of resources of type k. Our Hybrid algorithm has two parameters, the number of iterations without improvement and the value α, where α ∈ (0, 1) parameter controls the greediness or randomness of the greedy procedure. The α values considered for the instances are 0.3, 0.5 and 0.7 (column 4). The number of iterations without improvement is 10 times for all instances. For each instance, the optimal solution (OptSol) is obtained with FICO-XPRESS optimization suite that uses Branch and Cut (B&C). For some instances it is impossible to find the optimal solution, then the reported solutions correspond to

the best solutions found (BestKnown) within a time limit of 3600 seconds. In (8) BestKnown denotes the best solution found after a time limit 3600 seconds or optimal solution OptSol (if the optimal solution is found), solved with B&C (using FICO Xpress-SLP ). The average percentage deviation (Gap) of the optimal solution (**) or best known solution(*) with respect to GRASP algorithm is shown in column 5, where Gap, (OptSol ) or (BestKnown) − GRASP solution × 100 Gap = (OptSol ) or (BestKnown) (8) Finally, column 6 show the average percentage deviation (Gap) of the optimal solution (**) or best known solution(*) with respect to Hybrid Algorithm. Where Gap, Gap =

(OptSol ) or (BestKnown) − HybridSol × 100 (9) (OptSol ) or (BestKnown) TABLE I GRASP AND H YBRID ALGORITHM RESULTS

Instance 1

J 2000

Ik 400

2

3000

600

3

5000

1000

4

8000

1600

5

9000

1800

6

10000

2000

Alpha value 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7

Gap (GRASP) 3.4836** 2.9276** 3.9892** 2.5315** 2.5315** 1.8977** 2.7874** 2.6271** 2.8930** 1.4398* 2.5379* 2.4009* 2.8710* 1.5634* 2.6681* 1.4746* 1.4782* 1.8840*

Gap(HybridAlg) 1.4152** 1.7739** 1.4210** 0.7935** 0.3142 ** 1.3402** 1.7242** 0.9010** 1.8531** 0.9152* 0.5412* 0.8438* 1.6136* 1.4872* 1.8993* 1.2552* 1.4746* 1.0388*

We make the following observations about the experiment: The table above shows that the maximum error is 3.98 with the GRASP algorithm (with an α = 0.7) while the maximum error for the proposed Hybrid algorithm is 1.89 (with an α = 0.7). We obtained good quality solutions (with both algorithms) and execution times were at most 4 minutes (Hybrid algorithm). The proposed Hybrid algorithm is good in time and quality of solutions. As can be observed, the proposed GRASP heuristic found tight solutions for several instances. Table I shows that the Hybrid Algorithm obtains better solutions than the GRASP. VI. C ONCLUSIONS In this paper, we have studied a variant of the Maximal Covering Location Problem applied to forest fires. A mathematical model is proposed for the location problem of forest fire fighting resources. The mathematical model is a binary mathematical programming model. The exact solution method used is the branch and bound. A hybrid algorithm for solving this problem was proposed. The quality of the heuristic solutions can be measured with an upper bound produced by best or optimal solutions found with a commercial software.

Computational results in generated test problems, indicate that the method finds solutions of very good quality. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support from the Mexican National Council of Science and Technology, CONACyT. R EFERENCES [1] S. Chanta, M. E. Mayorga, and M. E. Kurz, “The minimum p-envy location problem: a new model for equitable distribution of emergency resources,” IIE Transactions on Healthcare Systems Engineering, vol. 1, no. 2, pp. 178–187, 2011. [2] V. Bennett, D. Eaton, and R. Church, “Selecting sites for rural health workers,” Social Science, vol. 2, no. 16, pp. 63–72, 1982. [3] D. Eaton, M. Daskin, B. Bulloch, and G. Jansma, “Determining medical service vehicle deployment in austin, texas,” Interfaces, pp. 96–108, 1985. [4] D. Sweeney, R. Mairose, and R. Martin, “Strategic planning in bank location,” in International Conference on Electronics, AIDS, Ed., 1979. [5] C. Chung, “Recent applications of the maximal covering location planning (m.c.l.p.) model,” Journal of the European Operational Research Society, vol. 37, no. 2, pp. 735–746, 1986. [6] D. L. Martell, “Operational research and forest fire management resource deployment,” in Operational Research ’90: Selected Papers from the Twelfth IFORS International Conference on Operational Research, H. E. Bradley, Ed. Oxford, United Kingdom: Pergamon Press, 1990, ch. 5, pp. 229–238. [7] R. A. Gonzalez-Gonzalez, V. Alarcon-Aquino, O. Starostenko, R. RosasRomero, J. M. Ramirez-Cortes, and J. Rodriguez-Asomoza, “WaveletBased Smoke Detection in Outdoor Video Sequences,” in 53rd IEEE Midwest Symposium on Circuits and Systems MWSCAS 2010, Seattle, Washington, USA, August 2010. [8] M. A. Finney, J. D. Cohen, S. S. McAllister, and W. M. Jolly, “On the need for a theory of wildland fire spread,” International Journal of Wildland Fire, 2012. [9] J. Gonz´alez-Olabarria and T. Pukkala, “Integrating fire risk considerations in landscape-level forest planning,” Forest Ecology and Management, vol. 206, no. 2, pp. 278–287, 2011. [10] X. Dong, D. Limin, S. Guofan, T. Lei, and W. Hui, “Forest fire risk zone mapping from satellite images and GIS for Baihe Forestry Bureau, Jilin, China,” Journal of Forestry Research, vol. 16, no. 3, pp. 169–174, 2005. [11] D. L. Martell, E. Gunn, and A. Weintraub, “Forest management challenges for operational researchers,” European Journal of Operational Research, vol. 104, no. 1, pp. 1–17, 1998. [12] J. P. Minas, J. W. Hearne, and J. W. Handmer, “A review of operations research methods applicable to wildfire management,” International Journal of Wildland Fire, vol. 3, pp. 189–196, 2011. [13] G. H. Donovan and D. B. Rideout, “An integer programming model to optimize resource allocation for wildfire containment,” Forest Science, vol. 49, no. 2, pp. 331–335, 2003. [14] I. T. Loane and J. S. Gould, Aerial Suppression of Bushfires. Camberra, Australia: National Bushfire Research Unit, 1986. [15] M. Dimopoulou and I. Giannikos, “Spatial optimization of resources deployment for forest-fire management,” International Transactions in Operational Reseach, vol. 8, no. 3, pp. 523–534, 2001. [16] R. Church and C. ReVelle, “The maximal covering location model,” Papers of the Regional Science Association, vol. 32, no. 1, pp. 101– 118, 1974. [17] G. Dantzig, “Reminiscences about the origins of linear programming,” Operations Research Letters, vol. 1, no. 2, pp. 43–48, 1982. [18] T. Feo and M. Resende, “Greedy randomized adaptive search procedures,” Journal of Global Optimization, vol. 6, pp. 109–134, 1986. [19] M. Resende and T. Feo, “A probabilistic heuristic for a computationally difficult set covering problem,” Journal of Global Optimization, vol. 8, pp. 67–71, 1989. [20] M. Resende, “Computing approximate solutions of the maximum covering problem with grasp,” Journal of Heuristics, vol. 4, pp. 161–177, 1998.