Determining Fire Station Location Using Convex Hull
I Putu Eka N. Kencana1, G.A. Purnawa2and Putu H.Gunawan3
1Department of Mathematics, Udayana University Kampus Bukit Jimbaran, Badung-Bali (Indonesia);
Email : [email protected]
2Department of Mathematics, Udayana University Kampus Bukit Jimbaran, Badung-Bali (Indonesia);
Email : [email protected]
3Universitée Paris-Est
LAMA UMR8050, F-77454, Marne-la-Vallée (France);
Email : [email protected]
ABSTRACT
The aim of this paper is to determine a location of fire station in South Kuta using convex hull and Dijkstra algorithm. The location was obtained using polygon convex approximation, where centroid of polygon is chosen as the best candidate. Housing coordinate points were used as sample points to construct convex hull using Graham Scan’s. The centroid of convex hull is the best candidate point to place the station in order to make Euclidean distances can be minimized.
Further, to minimize response time, Dijkstra algorithm used to find shortest route from candidate point of fire station to every fire-prone points. The obtained location will be compared with the location of existing fire station to examine its performance.
Keywords:Convex hull; Dijkstra algorithm; fire station; Kuta; minimax facility location.
Mathematics Subject Classification: 68R10, 97K30
1. INTRODUCTION
Siting an emergency facility such as fire station, can be modeled with minimax facility location problem [1]. Minimax facility location problem is a siting facility model where its goal was directed to minimize the maximum distance between new facility with its demand points. This model be appropriate used in determining new fire stations location because of the maximum distance from fire station location to demand points could be minimized. It aimed to maximize services if fire happened regarding to maximum response time allowed is less than 15 minutes [2].
Some researches about fire station location analysis had been conducted. For example, Arogundade et al. applied balas additive algorithm to determine fire stations location and emergency facility in Nigeria [3], Liu et al. determine fire stations location using Geographic Information Systems (GIS) and Ant algorithm [4], and Yang et al. developed fuzzy multi-objective programming model to determine optimal fire stations location through genetic algorithms [5].
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In this paper, we determined fire station location by adopting different approach. Using computational geometry and weighted graph concepts, we built model to find the best fire station location candidate in South Kuta considering two factors, namely distance and response time. Both factors were crucial in siting emergency facility location such as fire station [6]. Geometrically, every fire-prone locations assumed as points of a convex polygon. The centroid of convex polygon can be viewed as a good fire station location considering the maximum distance from this location to the farthest point be minimized. We constructed convex polygon using Graham Scan algorithm. This algorithm had been applied for constructing convex hull to solve various problems such as pattern recognitions, GIS, robotic, and image processing [7].
The remainder of this paper is organized as follows. Literature review presents theories that associated and support this paper problem. The computational result and comparison of both fire station location are present in result and discussion part. Finally, we present the conclusion of this paper.
2. LITERATURE REVIEW
2.1. Convex Hull
Given S is a set of points in Ըଶ. S is called convex if ሺݔǡ ݕሻ א ܵ, then segment ݔݕതതത ك . Convex Hull of points set S element is smallest convex set containing S (Fig. 1).
Figure 1. (a) Convex Hull; (b) Non-convex Hull
2.2. Graham Scan Algorithm
Graham Scan algorithm was introduced by American mathematician Ronald Lewis Graham in 1972 and used to determine convex hull of finite point sets in Ըଶ with complexity time is ܱሺ݊ ݊ሻ, where ݊ is number of points entered. According to [8], the first step to determine convex hull of finite points set is to select the starting sorting point. Selected origin point is point with smallest ordinate to the point (0,0). If there were more than one point with smallest ordinate, then the selected point is the point with the largest abscissa or rightmost point. The sorting origin point denoted by and the most opposite clockwise point denoted by ିଵ.
The next step is to sort points in ascending order by its counter clockwise angle formed by horizontal line through. If point and collinear throughp0, then be on the order of precedence over . Points that had been sequenced denoted by ǡ ଵǡ ǥ ǡ ିଵ where is sorting origin and ିଵas the most counter clockwise point. The sequenced points be maintained in the stack S. Because and ଵ
form an extreme angle, then ଵbe on the hull. Therefore, the initial condition that there are at least two points on the stack were met, namely ܵ ൌ ሺଵǡ ሻ where ଵ on the top.
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obtained from Fire Department of Badung Regency. Data consist of population density and number of fires by object. Housing was chosen as a population of points and 15 points were taken randomly.
Fifteenth of this data represent all of housing point in South Kuta.
The coordinate of sample points were obtained from Google Earth software. These coordinates using geographic coordinate in Universe Transvers Mercator (UTM) form. These coordinate points will be transformed into Cartesian coordinate by assuming equator line as x axis and central meridian as y axis. Cartesian coordinate of sample points used to construct convex hull with Graham Scan algorithm. After the convex hull constructed, fire station location is determined by finding centroid of convex hull.
To optimize response time, routes of fire truck will be minimized using Dijkstra algorithm. All routes to every sample point is identified based on wide of road and road condition. All routes will be modeled into graph to get shortest path from fire station location to every sample point.
4. RESULT AND DISCUSSION
4.1. Transformation of UTM to Cartesian Coordinates
South Kuta is located in the south of equator line and west of central meridian. To convert the UTM coordinate system to Cartesian system, we set the equator line an artificial value of 10.000.000 meters, decrease from north to south; and set central meridian an artificial value of 500.000 meters, decrease from east to west. Based on these values, the absisca (x) of a site in South Kuta was obtained by subtracting 10.000.000 meters with northing coordinate in UTM and the ordinate (y) obtained by subtracting 500.000 meters with easting coordinate [11]. Table 1 show UTM and Cartesian coordinates of sample points of housing in South Kuta.
Table 1: UTM and Cartesian Coordinates of Sample Points
Point
(pi) Housing Name
UTM Coordinates Cartesian Coordinates* Zone Easting
(m E)
Northing
(m S) X (m) Y (m) 1 Perumahan Kosala Jimbaran 50 L 300316 9029295 -199684 -970705 2 Perumahan Puri Mumbul Permai 50 L 301261 9028385 -198739 -971615
3 Pondok Bukit 50 L 297530 9027788 -202470 -972212
4 Perum Taman Penta 50 L 297627 9027353 -202373 -972647
5 Perum Taman Mulia 50 L 299355 9028849 -200645 -971151
6 Rumah Kos Jimbaran 50 L 298168 9026225 -201832 -973775 7 Perumahan Taman Sakura 50 L 296145 9027160 -203855 -972840
8 Perum Wisma Nusa 50 L 303137 9026526 -196863 -973474
9 Perum Puri Madani 50 L 302931 9026440 -197069 -973560
10 Perum Pondok Kampial Permai 50 L 302216 9026016 -197784 -973984
11 Bualu Indah 50 L 304519 9026563 -195481 -973437
12 Perumahan Ungasan Residence 50 L 298442 9022968 -201558 -977032
13 Tanjung Benoa 50 L 304321 9031675 -195679 -968325
14 Perum Permata Nusa Dua 50 L 303838 9026405 -196162 -973595
15 Labuan Sait 50 L 293898 9023115 -206102 -976885
* South Kuta located at the fourth quadrant of Cartesian System
4.2. Constructing Convex Hull
Using Matlab software, sample points be plotted into Cartesian coordinates like figure 3. The initial step in constructing convex hull is to determine the sample point with smallest ordinate. This point will be used as sorting origin point p0. Sample point with smallest ordinate is ൌ ሺെʹͲͳͷͷͺǡ െͻ͵Ͳ͵ʹሻthat given star sign as depicted in fig. 3 (scaled in km).
Figure 3. Position of Sample Points at Cartesian System
Once point p0 is determined, the next step is to sort the rest of sample points in ascending order.
Sorting isbased on the angle formed (counter clockwise) to horizontal line through p0 as depicted in fig.
4.
Figure 4. Sorting points based on the angle formed by counter clockwise to horizontal line through p0
Applying Graham’s algorithm, the final stack for constructed convex hull was ܵ ൌ ሺଵସǡ ଵଷǡ ǡ ଵǡ ሻ and the centroid of this convex hull was ൌ ሺെʹͲͲͳͳͷǡ െͻ͵͵Ͳሻ as depicted at fig. 5 with small circle symbol.
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4.3. Co rmine the pe fire station g fire accide esponse time
d to be 56.4 ire Fighting ess. Shortes ph. Main ro routes impa marks with i will be model the interse able by truck
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Figure 5. Fin mparison b erformance o location at B ents. Respo e is one rou 4 km/hour (b
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Using Dijkstra algorithm, shortest route is showed in fig.7. The shortest route is A – B – E – F with the length as much as 2.74 km, so that response time from fire station location by using convex hull to Rumah Kos Jimbaran is 2.91 minutes. Response time for another sample points could be obtained with same procedure and the comparison of both response time to fight fire accidents showed in Table 2.
Figure 7. The Shortest Path to Rumah Kos Jimbaran Table 2: The Comparison of Route Length and Response Time
Point
(pi) Housing Name
Shortest Route from Fire Station (km)
Response Time from Fire Station (km) Using
CH
Existing Location
Using CH
Existing Location
1 Perumahan Kosala Jimbaran 5.57 8.79 5.93 9.35
2 Perumahan Puri Mumbul Permai 6.28 9.50 6.68 10.11
3 Pondok Bukit 4.50 5.97 4.78 6.35
4 Perum Taman Penta 3.63 5.10 3.86 5.42
5 Perum Taman Mulia 4.76 7.40 5.06 7.87
6 Rumah Kos Jimbaran 2.74 4.48 2.91 4.76
7 Perumahan Taman Sakura 5.97 6.09 6.35 6.47
8 Perum Wisma Nusa 9.58 10.57 10.19 11.25
9 Perum Puri Madani 9.27 10.26 9.86 10.92
10 Perum Pondok Kampial Permai 8.38 9.50 8.92 10.11
11 Bualu Indah 10.23 12.24 10.88 13.02
12 Perumahan Ungasan Residence 5.05 5.09 5.37 5.42
13 Tanjung Benoa 13.93 16.96 14.82 18.04
14 Perum Permata Nusa Dua 10.06 11.79 10.70 12.54
15 Labuan Sait 9.04 3.20 9.62 3.41
The calculation showed, for the farthest sample point i.e. Tanjung Benoa Housing, suggested fire station location had maximum response time as much as 14.82 minutes meanwhile this time counted from existing fire station at BPG was 18.04 minutes. From these figures, suggested fire station location obtained using convex hull is better than existing fire station in BPG.
5. CONCLUSION AND SUGGESTION
We concluded convex hull can be applied to determine fire station location. The suggested location as the centroid of the convex hull outperforms the existing location based on response time needed to fight fire accidents. In determining the optimal location for fire station, this paper only considering Euclidean distances. To get better location, additional factors should be considered such as traffic density, length and wide of roads, and population density. Some extension to this research could be done in the future. For instance, including population and traffic density variables to find the centroid is suggested.
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[7] O'Rourke, J., 1997. COMPUTATIONAL GEOMETRY IN C. 2nd ed. Massachusetts: Cambridge University Press.
[8] Devadoss, S.L. & O'Rourke, J., 2011. Discrete and Computational Geometry. New Jersey:
Princeton University Press.
[9] Bourke, P., 1988. Polygon Area and Centroid. [Online]
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http://www.badungkab.bps.go.id/kecamatan_dalam_angka/2013/kuta_selatan [Accessed 4 Januari 2013].
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