AbstractIn this paper, an improved quadratic programing formulation for the solution of unweighted Euclidean 1-center location problem is presented. The original quadratic program is proposed by Nair and Chandrasekaran in 1971. Besides, they proposed a geometric approach for problem solving. Then, they concluded that the geometric approach is more efficient than the quadratic program. This conclusion is true only when all decision variables are treated as nonnegative variables. To improve the quadratic program, one of those variables should be an unrestricted variable as it is presented here. Numerically we proved that the improved quadratic program leads to the optimal solution of the problem in parts of second regardless of the size of the problem. Moreover, constrained version of the problem is solved optimally via the improved quadratic program in parts of second

Al-Zahrani, Khalid

El-Tamimi, Abdul Aziz

Elsevier - Publisher Connector 

Journal of King Saud University – Engineering Sciences (2013) 25, 161–165King Saud University
Journal of King Saud University – Engineering Sciences
www.ksu.edu.sa
www.sciencedirect.comSHORT COMMUNICATIONAn improved quadratic program for unweighted Euclidean
1-center location problemAbdul Aziz El-Tamimi a, Khalid Al-Zahrani b,*a Manufacturing System Engineering, PO Box 800, Riyadh 11421, Saudi Arabia
b Manufacturing System Engineering, Department of Industrial Engineering, King Saud University, Saudi ArabiaReceived 19 September 2011; accepted 29 April 2012
Available online 9 May 2012*
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Location;
1-Center;
Circle covering;
Quadratic programCorresponding author. Tel.
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18-3639 ª 2012 King Saud U
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es.2012.0Abstract In this paper, an improved quadratic programing formulation for the solution of
unweighted Euclidean 1-center location problem is presented. The original quadratic program is
proposed by Nair and Chandrasekaran in 1971. Besides, they proposed a geometric approach
for problem solving. Then, they concluded that the geometric approach is more efﬁcient than the
quadratic program. This conclusion is true only when all decision variables are treated as nonneg-
ative variables. To improve the quadratic program, one of those variables should be an unrestricted
variable as it is presented here. Numerically we proved that the improved quadratic program leads
to the optimal solution of the problem in parts of second regardless of the size of the problem.
Moreover, constrained version of the problem is solved optimally via the improved quadratic pro-
gram in parts of second.
ª 2012 King Saud University. Production and hosting by Elsevier B.V. All rights reserved.1. Introduction
1-Center Euclidean location problem is introduced originally
by Sylvester (1857). The problem involves enclosing m known
points in the plane within a circle of minimum radius. Con-
trary to what a person might think, the problem cannot be
solved by vision (or at least no one has yet been able to do
so) (Francis et al., 1992).
This problem is also known as the circle covering problem,
minimum spanning circle, smallest enclosing circle (or disk)08622236.
(A.A. El-Tamimi), Safe4k@
Saud University.
g by Elsevier
. Production and hosting by Elsev
4.003and single facility minimax location problem. In the circle cov-
ering problem as shown in Fig. 1, it is wished to locate a new
facility with respect to m demand points so as to minimize the
maximum Euclidean distance from the new facility to the de-
mand points. Thus, the objective is to minimize the function
g(x,y) deﬁned by:
gðx; yÞ ¼ maxf½ðx aiÞ2þ ðy biÞ21=2 : 1  i  mg ð1Þ
where (x,y) are the new facility coordinates; (ai,bi) are the de-
mand points (Pi) coordinates i= 1, . . .,m.
A problem equivalent to minimizing g(x,y) is to minimize
the maximum Euclidean distance (Z) as follows (Farahani
and Hekmatfar, 2009):
Min Z
Subject to : ½ðx aiÞ2 þ ðy biÞ21=2  Z i ¼ 1; . . . ;m ð2Þier B.V. All rights reserved.
Figure 1 Circle covering problem example.
162 A.A. El-Tamimi, K. Al-ZahraniThe circle covering problem may be of interest in locating a
radio transmitter, a radio receiver, radar station, hospital for
emergency cases, ﬁre station and police ofﬁce. Also, the prob-
lem of stationing a helicopter so as to minimize the maximum
time for it to respond to an emergency at any one of m sites is
closely related to this problem.
2. Review of literature
Many solution approaches are suggested in the literature to
solve the circle covering problem as shown in Table 1.
Nair and Chandrasekaran (1971) proposed geometric ap-
proach and quadratic program to solve the problem. Elizinga
and Hearn (1972) proposed geometric method for the solution
of the problem. Unweighted Euclidean and rectilinear dis-
tances are considered. The complexity of the algorithm is
O(n2), where n is the number of demand points. Drezner and
Wesolowsky (1980) presented a fast iterative method for locat-
ing one center on the plane. Weighted and unweighted dis-
tances are considered. The general lp-norm (pP 1) is used as
distance measure. A 3000 demand point problem in Euclidean
distance is solved in parts of second. Chandrasekaran (1982)
presented a polynomial algorithm to solve the weighted
Euclidean 1-center problem. The algorithm is proposed to min-
imize the ratio of convex quadratic and an afﬁne function offer
a polynomial set. The complexity of the algorithm is polyno-
mial in the dimension of space. Megiddo (1983) presented aTable 1 Literature summary.
Authors Year Distance
Nair and
Chandrasekaran
1971 Unweighted Euclidean
Elzinga and Hearn 1972 Unweighted Euclidean and Rectiline
Drezner and
Wesolowsky
1980 Weighted and unweighted lp-norm
Chandrasekaran 1982 Weighted Euclidean
Megiddo 1983 Unweighted Euclidean, weighted rec
tree network
Datta 1996 Unweighted Euclidean
Ohsawa and Imai 1997 Unweighted Euclidean
Matsutomi and Ishii 1998 A-distance
Das et al. 1999 Unweighted Euclidean on a hemisph
Li et al. 2002 Unweighted Euclidean
Brimberg and
Wesolowsky
2002 Unweighted Euclidean with area fac
Roy et al. 2009 Unweighted Euclidean with constralinear time algorithm to solve the problem. Datta (1996) pro-
posed an algorithm based on the concept of self-organizing
neural networks to solve the problem. The worst-case com-
plexity of the proposed algorithm is O(logn) where n is the
number of demand points. Ohsawa and Imai (1997) presented
a procedure to construct contour lines and compute the area of
the region where the objective function value is equal or less
than a constant value for the problem based on the farthest
point Voronoi diagram. Matsutomi and Ishii (1998) proposed
a solution procedure to solve the problem when A-distance is
considered. The procedure is an extending of the geometric ap-
proach proposed by Elizinga and Hearn (1972) to A-distance
case. Then they applied the procedure to the location of an
ambulance service station in an area. Das et al. (1999) pro-
posed an algorithm to solve the problem when the demand
points are spread over a hemisphere. The algorithm is based
on geometry and having a time complexity O(n2) where n is
the number of demand points. Li et al. (2002) considered
two fuzzy versions of the circle covering problem when the
locations of points are not precise but fuzzy. Polynomial algo-
rithms are proposed for both versions. Brimberg and Weso-
lowsky (2002) formulated the problem on the continuous
plane where the demand points and service center may be rep-
resented by areas instead of points. The distance function mea-
sures the shortest distance between any point on the service
center and any point in the demand area. Also, a general meth-
odology for optimization was developed and can lead to efﬁ-
cient solution methods. Roy et al. (2009) proposed a
heuristic algorithm to solve the circle covering problem, where
the center is constrained to lie on a query line segment. The
time complexity of the algorithm is O(log2n) where, n is the
number of demand points.3. Quadratic programing model
Nair and Chandrasekaran (1971) proposed a quadratic pro-
graming formulation for the solution of the problem. The
problem is converted into an equivalent quadratic programing
problem as follows:Method/Formula
Geometrical approach
Quadratic program
ar Geometrical approach
Heuristic approach
Polynomial algorithm
tilinear and weighted Heuristic approach
Heuristic approach based on Neural
network
Geometric approach to construct contour
lines
Geometric approach
ere Heuristic approach
Heuristic approach
ility General methodology
ints Heuristic approach
Figure 3 IQP vs. QP solution, m= 200.
An improved quadratic program for unweighted Euclidean 1-center location problem 163Let the coordinates of the points Pi be denoted by (ai,bi),
i= 1, . . .,m and those of the point P by (x,y) (Li et al., 2002):
Define dðx; yÞ ¼ max dðP;PiÞ; i ¼ 1; . . . ;m:
Then dðx; yÞP dðP;PiÞ; i ¼ 1; . . . ;m or equivalently;
d2ðx; yÞP ðx aiÞ2 þ ðy biÞ2; i ¼ 1; . . . ;m: ð3Þ
By deﬁning new variable k= d2  x2  y2, the problem is
reduced to the following quadratic programing model:
Minimize fðk; x; yÞ ¼ kþ x2 þ y2 ð4Þ
Subject to : 2aixþ 2biyþ k  a2i þ b2i ; i ¼ 1; . . . ;m ð5Þ
Nair and Chandrasekaran (1971) did not attempt to compare
their two proposed methods. They stated that, in all practical
problems the geometric method will be done manually while
the quadratic program will be solved on a computer. When
the two methods require such different means a comparison
of computational time is not meaningful. However, based on
their prior experience with quadratic programs, they strongly
believe that the geometric method will be more efﬁcient and
it is very simple.
One can notice that this model is not complete, since,
bounded constraints i.e., nonnegative constraints, are not de-
ﬁned. Thus, when it is wanted to solve this model, we will as-
sume that all deﬁned variables i.e., x, y, and k, are nonnegative
variables. In this case, Nair and Chandrasekaran’s conclusions
about the quadratic model will be true. If we assume that ﬁrst
quarter space will be considered, variables x and y will be non-
negative. The third variable k should be investigated as
follows:
The objective function i.e., f(k,x,y) = k+ x2 + y2, is to
minimize the maximum Euclidean distance. Hence, the dis-
tance d will never be negative. In addition, if x= y= 0, then
k must be positive. Oppositely, if the optimal solution of the
problem for example is x= 4, y= 5 and d= 2, then if we
substitute these values in the equation k= d2  x2  y2, we
will ﬁnd that k= (2)2  (4)2  (5)2 = 4 – 16  25 = 37Figure 2 IQP vs. QP solution, m= 10.which is a negative value. Thus, variable k is an unrestricted
variable.
4. Numerical results
To ﬁnd out how effective is the improved quadratic program
model, i.e., k is the unrestricted variable, random coordinate
data sets for 10 and 200 points, representing the demand
points, is generated from a uniform distribution U(0,1000)
and third data set for 500 points is generated from a uniform
distribution between U(0,1000) by ﬁrst generating x-coordi-
nate values then generating y-coordinate values.Figure 4 IQP vs. QP solution, m= 500.
Table 2 IQP vs. QP results summary.
Number of Demand points (m) Solution IQP QP
10 x-Coordinate 51.33886 49.40251
y-Coordinate 58.33431 46.83893
Circle radius 59.1794 68.0771
Iterations 10 8
200 x-Coordinate 51.17656 49.25605
y-Coordinate 48.92306 48.87072
Circle radius 68.0492 69.3866
Iterations 9 9
500 x-Coordinate 512.5 482.5
y-Coordinate 506.5 485.5
Circle radius 648.4724 684.4827
Iterations 15 11
Table 4 CPU time in seconds for IQP and nonlinear program.
Number of demand points (m) IQP Nonlinear program
500 <1 43
1000 <1 172
3000 <1 1085
5000 <1 4240
Table 3 IQP vs. D&W algorithm (Drezner and Wesolowsky algorithm).
Number of demand points (m) Solution IQP D&W algorithm
500 x-Coodinate 483.9797 483.9797
y-Coordinate 497.8540 497.8540
Circle radius 684.5700 684.5700
1000 x-Coodinate 498 498
y-Coordinate 507.5 507.5
Circle radius 687.6767 687.6767
3000 x-Coodinate 50.5 50.5
y-Coordinate 50 50
Circle radius 69.6509 69.6509
5000 x-Coodinate 503.4948 503.4948
y-Coordinate 506.9365 506.9365
Circle radius 694.0741 694.0741
Figure 5 Constrained 1-center, m= 10.
164 A.A. El-Tamimi, K. Al-ZahraniThe IQP (improved quadratic program) is compared to the
QP (original quadratic program) proposed by Nair and
Chandrasekaran to solve the unweighted Euclidean 1-center
location problem. IQP and QP are modeled in LINGO 11
(LINDO systems Inc.) then, the comparisons are run on
T7200 2 GHz with 2 GB RAM.
It is known that, in the optimal solution of the circle covering
problem, the circle is determined by two or three demand points
lying on its circumference (Francis et al., 1992). Regarding this
fact, an instance of 10 demand points is solved via both IQP and
QPas shown inFig. 2. Then, an instance of 200 demand points is
solved as shown in Fig. 3. Finally, instance of 500 demand
points is solved as shown in Fig. 4. It can be seen clearly that
IQP leads to the optimal solution of the problem in all cases
while, QP does not. The results are summarized in Table 2.
An improved quadratic program for unweighted Euclidean 1-center location problem 165Moreover, the IQP is compared to a fast heuristic algorithm
proposed by Drezner andWesolowsky (1980). D&W algorithm
(Drezner and Wesolowsky algorithm) is coded on MATLAB
(Mathwork Inc.) then, the comparisons are run. Large size
instance problems; i.e., 500, 1000, 3000 and 5000 points; are
generated from a uniform distribution U(0,1000). In all cases,
the solution values obtained by both methods were exactly sim-
ilar as shown in Table 3. However, the solutions are obtained in
parts of second via both methods.
Finally, the IQP is compared to the nonlinear program
(presented in Section 1) to solve constrained; i.e., the center
must or must not lie on a speciﬁc point; and unconstrained ver-
sion of the problem. The novelty of the IQP is due to the solu-
tion time and its ability to solve the constrained version of the
problem. Thus, optimal solutions of the problems are obtained
in parts of second. The nonlinear program leads to the optimal
solution in long CPU time; e.g. 5000 demand points are solved
optimally in more than 70 min. CPU times for IQP and nonlin-
ear program are shown in Table 4. In Fig. 5 the 10 demand
points instance solution when the center is constrained to lie
on a line having the equation x= y is illustrated.5. Conclusions
1-Center location problem is one of the best known location
problems. Geometric approaches and a quadratic program
are proposed in the literature to solve the problem. The
authors who proposed the quadratic program concluded that
the geometric approach is more efﬁcient than the quadratic
program. This conclusion is true only when all decision vari-
ables are treated as nonnegative variables in the quadratic pro-
gram, which is not right. One of those variables should be an
unrestricted variable as it is presented in the improved qua-
dratic program. The comparison between the improved qua-
dratic program and the original one shows that the improved
quadratic program leads to the optimal solution while, the ori-
ginal one does not. Moreover, the improved quadratic pro-
gram is compared to a fast heuristic algorithm proposed by
Drezner and Wesolowsky (1980). Numerically we proved that
the improved quadratic program leads to the optimal solution
of the problem in parts of second regardless of the size of theproblem. In addition, constrained version of the problem is
solved optimally via the IQP in parts of second.
References
Brimberg, J., Wesolowsky, G.O., 2002. Locating facilities by minimax
relative to closest points of demand areas. Computers and
Operations Research 29, 625–636.
Chandrasekaran, R., 1982. The weighted Euclidean 1-center problem.
Operations Research Letters 1 (3), 111–112.
Das, P., Chakraborty, N.R., Chaudhuri, P.K., 1999. A polynomial
time algorithm for a hemispherical minimax location problem.
Operations Research Letters 24, 57–63.
Datta, A., 1996. Computing minimum spanning circle by self-
organization. Neurocomputing 13, 75–83.
Drezner, Z., Wesolowsky, G.O., 1980. Single facility lp-distance min–
max location. SIAM Journal of Algebraic and Discrete Mathe-
matics 1, 315–321.
Elizinga, J., Hearn, D.W., 1972. Geometrical solution to some
minimax location problems. Transport Science 6, 379–394.
Farahani, R.Z., Hekmatfar, M., 2009. Facility Location Concepts,
Models, Algorithms and Case Studies. Springer, Berlin.
Francis, R.L., McGinnis, L.F., White, J.A., 1992. Facility Layout and
Location: An Analytical Approach, second ed. Prentice Hall, New
Jersey.
Li, L., Kabadi, S.N., Nair, K.P.K., 2002. Fuzzy versions of the
covering circle problem. European Journal of Operational
Research 137, 93–109.
Matsutomi, T., Ishii, H., 1998. Minimax location problem with A-
distance. Journal of the Operations Research Society of Japan 41
(2), 181–195.
Megiddo, N., 1983. Linear-time algorithms for linear programming in
R3 and related problems. SIAM Journal on Computing 12, 759–
776.
Nair, K.P.K., Chandrasekaran, R., 1971. Optimal location of a single
service center of certain types. Naval Research Logistics Quarterly
18, 503–509.
Ohsawa, Y., Imai, A., 1997. Degree of locational freedom in a single
facility Euclidean minimax location model. Location Science 5 (1),
29–45.
Roy, S., Karmakar, A., Das, S., Nandy, S.C., 2009. Constrained
minimum enclosing circle with center on a query line segment.
Computational Geometry 42, 632–638.
Sylvester, J.J., 1857. A question in the geometry of situation. Quarterly
Journal of Mathematics, 1–79.


An improved quadratic program for unweighted Euclidean 1-center location problem 

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An improved quadratic program for unweighted Euclidean 1-center location problem

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