Ambulance Service Area Placing with Overlapped
Efficiency Indexes
Dah-Ming Shiah
Yung-Tang Shen
Shu-Wen Chen
Instructor of Department of Nursing/
Graduate School of Architecture and
Urban Design
Central Taiwan University of Science
and Technology/ Chaoyang
University of Technology
Taichung, R. O. C.
Graduate School of Architecture and
Urban Design
Chaoyang University of Technology
Taichung, R. O. C.
Instructor of Department of Nursing
Central Taiwan University of Science and
Technology
Taichung, R. O. C.
Corresponding author:swchen@ctust.edu.tw
Abstract—In this study, we were able to create an automatic
overlapped placing of ambulance site using computer procedures
and linear programming. This placing scheme uses two efficiency
indexes Covered Rate and Overlapped Rate to evaluate the
performance while placing the service area into our study area.
The objective is to balance between covered rate and overlapped
rate with a given ratio. Two phases were suggested, first phase is
performed by honey cone search and minimum buffer search to
choose the location of next potential site. The second phase moves
the potential site so that the objective is met. The service areas of
facilities are bounded by both acceptable traveling distance and
service capacity. This means that the shape and size of service area
changes while the facility point changes its location, which is
fundamentally different from Bin Packing or Placement problem.
Therefore, we have to extinct this study from them and referred as
Overlapped Placing. The benefit of using auto Overlapped Placing
is that based on the changing circumstances the alternative
placing results can be generated relatively fast and easy to satisfy
the need of deploying emergency service in a highly populated
urban area.
Keywords-component; Geography Information System (GIS);
Efficiency Index; Bin Packing; Placement.
I.
INTRODUCTION
The preliminary results and schemes of automatically
placing the ambulance service area into a give area are presented
in this paper. This study is part of an on going research
conducted by authors, although this paper presents part of
solutions, there are many questions and problems need to work
on. The results show that using different parameters in searching
procedure to fill up the study area will takes about 30 minutes in
average and 19 to 44 minutes in extreme cases in a regular PC
with 3GHz. CPU. However, there are some cases where no
result can be generated.
In the ambulance location/allocation studies the concern are
mostly on the linear programming models, whereas, how to
place or pack facilities and service area were rarely mentioned.
In this study, we suggested a heuristic method consist of two
efficiency indexes, Covered Rate and Overlapped Rate, to
evaluate the performance of placing the facility into the study
This study is partially sponsored by National Science Council.
area. The objective seeks to balance between covered rate and
overlapped rate with a given ratio. Two phases of auto placing
were suggested in this study, first phase is performed by honey
cone search and the second phase minimum buffer search to
choose the next possible site. The second phase moves the
potential site so that the objective is met. The procedures of auto
placement that we developed, is programmed by Delphi 2005
and used MapInfo as GIS operations. In the developing process,
we found that under overlapped consideration, it took O(n)
calculations to generate an acceptable result. Further, the
service areas are bounded by both acceptable traveling distance
and service capacity, as described by AACM. This means that
the shape and size of service area changes while the facility
point changes its location, which is fundamentally different
from Bin Packing or Placement problem in the operation
research study and we referred as Overlapped Placing. We
hope that by suggesting auto overlapped placing, where road
structures and population concentration were considered, can
replace the complex mathematical format of dynamic models in
ambulance location /allocation study. Another benefit is that
based on the fast changing conditions of ambulance service the
alternative placing results can be generated fast and easy by
using our procedure to satisfy the need of deploying emergency
service in a highly populated urban area.
II.
REVIEW
The location set covering model (LSCM) proposed by
Toregas et al. (1971) was one of the early ambulance model and
classified as deterministic model by Brotcorne[1]. Krarup[2]
also regarded the set covering, set partitioning, and set packing
models as the special cases of un-capacitated facility location
problem (UFLP) and were categorized into pull-objectives
models. The aim of the LSCM is to minimize the number of
ambulances needed to cover all demand points, under the
conditions that each ambulance site covers an area within the
travel distance of preset time. This model was illustrated by
binary linear programming but was unable to describe the detail
of how the locations were selected.
In the packing research, there are two types of twodimensional packing concerning heuristics according to Lodi[3],
off-line algorithms assumed that the algorithm has full
knowledge of the whole input and on-line algorithms which
pack each item as soon as it is encountered. Off-line algorithms
are mostly greedy and can be classified in two families (1)
one-phase algorithms directly pack the items into the finite bins;
(2) two-phase algorithms start by packing the items into a single
strip. Lodi [4] also defined packing problems into twodimensional bin packing problems and strip packing problems.
Both articles adopted integer linear programming and binary
linear programming to generate approximation algorithms,
lower bounds and exact algorithms. Bin packing provided
excellent solutions toward placing objects into desired area set,
however, due to assumption it can only placing rectangle shaped
object and no overlapping allowed, therefore, different from our
study.
Placement, also called layout or floor planning, is another
area of research where objects with given shapes are placing into
designated floor or area set. Placement were first used in very
large scale integration (VLSI) physical design and the aim is to
minimize the chip size and optimize the global interconnect
structure. Efficient, flexible and effective representation become
the indexes of evaluate the outcome of different models. The
non-slicing floorplan representations includes sequence pair
(SP), bounded-slice line grid (BSG), O-tree, B*-tree, corner
block list (CBL), and transitive closure graph (TCG) according
to Lin[5]. Although placement is an excellent method to solve
designing problem in VLSI but it is mostly heuristic and did not
provide the solution of placing objects into an empty set and also
did not allow overlapping of objects.
Perhaps most related research to our study is the irregular
shaped objects packing in two-dimensions. According to Chen
[6], there are less works done in arbitrary shapes in comparing
with bin packing. Chen [6] introduces the Boundary
Representation to solve irregular shaped objects packing
problems. It offers three ways to reach the local optimal,
namely shifting down-most, shifting left-most and rotation. He
also proposed three packing sequence pack object with largest
area first, pack object with smallest area first and pack objects in
random order. His study shows that by using pack object with
largest area first can come out with better performance. The
Boundary Representation is similar to what we are proposing
except that the ambulance service area is defined as convex hull
shape object, we do not know the shape of each object in
advance and objects can overlap with each other.
This study was originated from Ambulance Allocation
Capacity Model (AACM) developed by authors of this paper.
AACM is designed to solve the same problem as LSCM did but
adding the capacity and address point data set in ambulance
location model. The aim of AACM is to minimize the number of
ambulances and the service areas needed to cover all demand
address points and area also take into consideration the capacity
of the service area of each ambulance in order to make necessary
adjustments to the traveling distances.
The improvement from AACM is shown in table I.
However, the arrangements of all location sites are manually
selected and took authors days to come out with an acceptable
result. This slow and painful process provokes the study of this
paper.
TABLE I.
THE RESULT FROM AACM IN TAICHUNG CITY
Ambulance type
Sites
Ambulances #
Area
Coverage
Address
Point
Coverage
Covered
Actual
Covered
Actual
Current BLS & ALS
23
40
%
Overlap
62.89%
48.08%
49.24%
%
Overlap
148.02% 53.67%
68.90%
III.
AACM
41
41
%
Overlap
110.11%
17.58%
90.74%
%
Overlap
104.71%
10.98
93.21%
MTEHOD
The covered rate (0<=Cr<=0) and overlapped rate (0<=Lr
<=1) are inherited from AACM. The covered rate is defined as
the sum of all service areas within the study area (Sa). The
overlapped rate is defined by the study area subtract the areas
that covered by convex hulls and divided by areas that covered
by convex hulls. For i = {1,…,n} the sum of all service areas is
X=Σni=1 xi, then CrX and CrSa, also LrS. If we assume that
the increment of Cr and Lr are ΔCr and ΔLr then minimum of
|ΔCi –γΔLi| become the objective function to find a possible
outcome.
From discussion above, it is obvious that increasing the
covered rate the residents will get better ambulance service,
however, it will also cause the waste of valuable ambulance
resource. Intuitively, when covered rate gets close to 100% the
overlapped rate will increase dramatically, this is because the
nature of convex hull. A balance between covered rate and
overlapped rate can be calculated to estimate the performance of
each result. This performance estimation, we called it efficient
index, can be a constant or a variable depending on data sets and
requirements. In this study we assume it to be a constant.
The study area, Taichung city as illustrated in Fig. 2, is
located in the center of Taiwan. The data sets used in this study
include single line road segment map, city limit boundary map
and address point map. All of these maps are in GIS format and
in TWD97 coordinate system. Currently, Taichung city has total
population of 1,032,778 and 347,392 in household number at the
end of 2005 (http://www.tccg.gov.tw/sys/SM_theme?page=
42ccebbf, accessed 2006/03/14). According to the “Standard of
Equipments and Manpower of the Fire Department in
Municipality Under the Jurisdiction of the Central Government”
dictated by the Executive Yuan in Taiwan, there should be
66,667 persons per ambulance, studied by Chuang [8], and is
used as the preset capacity standard in this study. The average
ambulance traveling time is 4 minutes, including the 0.5 minutes
for dispatching commands and operation time studied by Chang
[9]. Therefore, the traveling time for each call for an ambulance
service is 3.5 minutes. The average travel speed of an ambulance
in the urban area is 24 kilometers per hour studied by Chuang[8]
and reaches a distance of 1.4 kilometers.
IV.
PROCEDURES
The first phase of generating potential sites is divided into
two stages. The first stage is the honey corn search which
assumes that by given a convex polygon there can be
approximately 6 convex polygons adjacent and surrounding it.
The second stage is the minimum buffer search which selects the
next site by calculating the shortest distance to the first site from
all potential sites. This process will repeat itself until no further
areas are available to select for next potential site. The second
phase is to adjust the potential site selected from phase one so
that the efficient index is met. The procedures are described as
following:
A. Give a starting site.
The starting site is given and is described in AACM by using
the highest gathering of address points. The choice of the first
location point is based on the assumption that if we locate the
highest demand gathering area first, then it will be easier to
arrange the less demand gathering areas (see ReVelle [10]
referring Eyster et al.[11]).
B. Honey corn search for next available site.
Intuitively, after the first convex polygon is identified we
can assume that a round of convex polygons can be placed
outside of the first convex polygon as the next potential sites in
the shape of honey corn (see Fig. 1a this is where the name
comes from). The program will search for the next highest
gathering point among these next potential sites and draw a new
convex polygon. This search will repeat until it reaches the fifth
convex polygon then move on to next procedure. The honey
corn search procedures are as following:
Honey corn search ()
Begin
Creates an external buffer from the first convex polygon
using given distance.
Calculate the address points falls within 1.4 Km.
distance of all the nodes of this new external buffer.
Find the node in the external buffer with highest address
point as the first selected site and draw the service area
convex polygon.
For i=3 to 5 Do Begin
Use the first site and the first selected site to find next
selected site clockwise and draw the new service area
convex polygon.
Move potential site to meet efficient index (refer to
procedures in D);
End ;
End ;
C. Minimum buffer serach for next available site.
The minimum buffer search draws an external buffer from
all the convex polygons selected before this site using an
estimated search radius to determine a new buffer polygon in
this borderline. Among this new borderline this procedure will
find a point closest to the first selected site and generate convex
polygon as the next potential site. The procedures to perform
minimum buffer search are as following:
Minimum buffer search ()
Begin
Creates an external buffer from all selected convex
polygons using estimated radius.
Find the node closest to the first selected site as the next
selected site.
End ;
D. Adjust selected site to satisfied efficient index.
In each selected site from procedure B and C, the selected
site will perform a shifting optimization by moving the selecting
site in different direction so that the covered rate and overlapped
rate are met with 2% of preset tolerance. The efficient index of
the percentage distance from potential site to the border of
adjacent convex polygon is 4.77% in circle shape (in our
program we use 5%) of the Euclidean distance (radius) of that
circle in order to reach the given 95% of overall covered rate.
Therefore, the aim is to move the potential site as close to 5% of
its radius as possible to all the overlapped convex polygons as
described in Fig. 1b.
Uncovered area
Limin
a
Li
Limax
Facility
Sitting
b
Overlapped
area
Figure 1. Honey corn shape and efficient index.
The Move potential site procedures are as following:
Move potential site ()
Begin
From the given selected site find the nearest two borders
of existed convex polygons.
For i=1 to 5 Do Begin
Draw convex polygon.
Check border overlapping to decide the moving
direction.
Calculate covered rate and overlapped rate.
If covered rate and overlapped rate are not met then
Begin
Adjust the overlapped area to meet the given
percentage for the first nearest polygon.
Adjust the overlapped area to meet the given
percentage for the second nearest polygon.
Else Break ;
End ;
End ;
Draw the new site and convex polygon.
End ;
The main procedures are listed below:
Draw convex polygon for the first site (A);
Honey corn search for next potential site (refer to
procedures in B);
For i=6 to infinity Do Begin
Minimum buffer search (refer to procedures in C);
Move potential site to meet efficient index (refer to
procedures in D);
End ;
V. RESULTS
The road single line map data set that used consists of 24,234
lines and the address point maps consist of 371,693 points. Fig.
3 gives a comparison of the results for different estimated radius
from 45% to 95% of preset traveling distance giving the same
covered rate and overlapped rate. We can clearly see that the
first six convex polygons including the starting polygon stay the
same in different radius. This is mostly because of the honey
corn procedures. When arranging polygons using minimum
buffer search the results varies.
The comparison of different selectable border percentages
and search radius versus time and selected sites are shown in
Table II. Please note that some results marked with (a) means
that no result can be generated by our program due to the spatial
circumstance at that location.
advantages, we believe that the procedures we introduced can
serve as a decision support system in location related operations.
Figure 2. Using 45%, 65%, 85% & 95% radius and 30% of selectable border
to estimat the next potentail site.
TABLE II.
Search radius
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
THE COMPARISON OF PERFORMANCE
Selectable border 10% of travel distance
Sites
Sec.
Sec./site covered rate Overlap rate
38
1779
46.8
77.33%
12.37%
36
1325
36.8
75.50%
12.27%
40
1609
40.2
79.92%
13.16%
44
1694
38.5
80.62%
15.17%
42
1852
44.1
80.92%
14.38%
41
1689
41.2
82.17%
15.34%
41
1715
41.8
82.10%
15.45%
48
1616
33.7
85.96%
17.13%
181 a
50
1934
38.7
83.77%
21.47%
51
1876
36.8
84.69%
19.73%
Selectable border 30% of travel distance
28
1484
53.0
62.89%
6.03%
33
1132
34.3
68.79%
7.00%
29
1322
45.6
69.04%
6.20%
35
1473
42.1
73.68%
7.92%
37
1598
43.2
74.32%
11.49%
37
2405
65.0
76.67%
10.33%
35
1585
45.3
76.78%
9.26%
41a
38
1636
43.1
78.30%
13.58%
45
1850
41.1
79.77%
19.13%
38
1803
47.4
80.15%
9.89%
Selectable border 50% of travel distance
25
2191
87.6
61.23%
3.85%
28
1229
43.9
63.16%
3.93%
31
2032
65.5
63.29%
13.23%
29
1684
58.1
61.91%
9.55%
28
1745
62.3
60.18%
6.59%
29
1280
44.1
64.85%
7.04%
30
1876
62.5
66.28%
5.65%
33
2125
64.4
70.72%
7.72%
37
2611
70.6
72.24%
13.09%
32
1881
58.8
70.62%
6.46%
36
2312
64.2
73.64%
9.27%
a. Unable to generate result and was stop by program.
VI. CONCLUSION
This study proves that automatic placing ambulance service
area and its selected site using fixed covered rate and overlapped
rate to generate acceptable result is achievable in most of the
cases that we tested. The Overlapped placing procedures that
proposed in this paper can give result in average of 30 minutes
which is considerable faster the manual operation. The
procedures accept changes of study area shape unlike in packing
problem the study area will have to be rectangle. With these
It is obvious that we try to use limited variables and simple
procedures to estimate a complicated spatial placing problem.
However, results show that 31 out of 33 cases were successful.
One of the improvements of this concern might be the learning
mechanism and the evaluation scheme to generate a better
estimation of next point. The shortest path and convex polygon
to generate service area are sensitive to road map data. When
long shape barrier block the connection of address points, it can
cause procedures to become unstable and lead to endless loop,
user should be aware. The moving potential site procedure
considers only nearest two sites may not suitable in some special
cases and often cause the fail of the whole procedures. This
study did not consider the holes inside of the study area but were
able to handle irregular polygon border shape by setting up
selectable border parameter. In these cases the moving potential
site procedure will stop at 5 tries preventing endless loop.
Therefore, there is no way to tell whether the result is
acceptable.
ACKNOWLEDGMENT
Thanks to National Science Council for providing part of the
funding and the universities for providing facilities.
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