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Some thoughts on generalized Weber location models
Appl. Math. Lett. Vol. 11, No. 1, pp. 121-126, 1998 Copyright(~)1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/98 $19.00 + 0.00
Pergamon
Pll: S0893-9659(97)00144-4
Some Thoughts on Generalized Weber Location Models M. J. KAISER Department of Industrial and Manufacturing Engineering Wichita State University, Wichita, KS 67260-0035, U.S.A. kaiser©ie, twsu. edu
(Received January 1996; revised and accepted November 1996) Abstract--The Weber and inverse Weber location problem is defined for a continuous onedimensional convex region in the plane and solved using constructive numerical techniques. It is conjectured t h a t the Weber functional for a continuous one-dimensional convex region is concave. The equivalence between the one-dimensionai inverse Weber model and a polar geometric optimization problem is demonstrated, and an alternative symbolic expression for the integral functional is described. Keywords--Constructive geometry, Generalized Weber problem, Geometric analysis, Continuous location models, Polar optimization.
The single-source discrete (zero-dimensional) Weber problem is the prototype location model. Given a set of points S m = {A1,... , A m } , where Ai = (xi,yi), i = 1 , . . . ,m, that form the set of cities, find the location of the facility to minimize the sum of the Euclidean distances from (x, y) to Ai = (xi, Yi), d((x, y), Ai), where
d ((x, y), Ai) = x / ( x - xi) 2 + (y - yi) 2. The zero-dimensional Weber location problem can thus be written as m
min ~
rid ((x, y), A i ) ,
(1)
i=1
where ti is a positive number (weight) associated with Ai, i = 1 , . . . ,m. The function defined by (1) is convex, and numerical methods (such as Weiszfeld's Algorithm or Newton's Method) to solve for this functional-type are well known [1, Chapter 4]. A zero-dimensional "inverse" Weber location model can be formulated as
min~
ti d ((x, y), Ai)"
(2)
i=1
This model admits a natural interpretation in terms of locating noxious facilities, but the form of the functional in this case is not convex, and thus, the solution point(s) are not readily characterized. Typeset by A2~S-TEX 121
122
M.J. KAISER
The zero-dimensional (0-D) "point set" location models motivate the following generalization. Consider a one-dimensional (l-D) "continuous" region defined as the boundary of a convex figure K (a "convex curve"). A convex polygon Pm with interior p o and boundary OPm will be used as the realization of K. Pm can be described in terms of its vertices as Pm = {A1,..., Am}, Ai -- (xi, Yi), i = 1 , . . . ,m, or in terms of the line segments L~ that compose its boundary as P~m = { L 1 , . . . ,Lm}, Li : aix+biy+ci = 0, i = 1 , . . . ,rn. Nonweighted Weber and inverse Weber location models for a continuous 1-D region defined from OPm can thus be written as sup
QeP&
fo
r ((Pro; Q), u) du,
du inf fo 2~ QeP& r((Pm;Q),u)'
(3)
(4)
where r((Pm; Q), u) is the radial function defined for u E S 1, the unit circle by
r((Pm;Q),u) = max{A _> 0 I Ax • Pm, u • $1}. The radial function is what we typically think of in terms of polar coordinates as '%racing out" the boundary of the figure. The radial function is a directed distance with respect to the point Q • p o and direction defined by u • S 1, and therefore, r((Pm; Q), u) is a function of Q and u • S 1. The radial function describes the radial distance from point Q to the boundary OPrn as a function of the counterclockwise angle u from the horizontal x-axis. The ray from Q • P m° in )
the direction defined by u, Q(u) intersects OP'm at point T(u). The radial function is simply the distance from Q to T(u), i.e., r((Pm;Q),u) = d(Q,T(u)), as u revolves about S t. For a small class of symmetric regions, the radial functional can be described in closed parametric form at the point of symmetry; in general, however, no simple representation exists for the radial function defined at any arbitrary point within a convex polygon. In general, r((Pm; Q), u) is a complicated functional dependent on the geometry of Pm which can only be solved numerically. As Q approaches the boundary of Pm, it is clear that r((Pm; Q), u) will approach zero, and the functional defined by (4) will subsequently "blow up". This implies that the natural formulation of (4) should be stated as a minimization problem. Model (3) represents (after the inclusion of a 1/2~r-term) the average radial value to the boundary OPm with respect to the reference point Q • Pr°n, and it is interesting that there is an extremal formulation associated with this problem. One possible interpretation of model (3) is as follows. Imagine defending a region where hostile forces frequently attack, and you want to locate your castle to maximize its average distance away from the boundary. The natural objective function to consider is described by (3) since a priori knowledge of the attack front is not known. An interpretation of model (4) is provided in Proposition 2.1. A "naive" solution methodology suggested by formulations (1),(2) does not work for the continuous Weber models. T h a t is, by selecting successively larger sets of points Ai • OPm, and then solving (1) or (2) "in the limit" would yield an unbounded functional value and no distinction between arbitrary interior points Q and Q' within Pro. The integration procedure is the key to avoid this infinity in the 1-D case and is a natural generalization to consider since it leads to "average" functional values. Recall [2, p. 620, Ex. 30]: the average value of r over the curve r = f(O), c~ < 0
f(O) dO.
In the case of a circle of radius a centered at the origin O, a = 0, f~ = 2~r, (r(f(O); 0)) = a. As the reference point Q moves within the curve, however, the value of the integral will change in a complicated fashion.
Weber Location Models
123
By analogy with the Euclidean distance function d((x, y), As), it is shown that for a convex polygon and fixed point Q E P°m, the radial functional r((Pm; Q), u) is a piecewise convex function of u E S 1. Since the distance function is convex, it is clear that the radial functional will be piecewise convex. This can also be shown as follows. PROPOSITION 1. For fixed Q e po, r( (Pm; Q), u) is a piecewise convex function. PROOF. Given point Q E F ° and the line segment Li of the polygonal representation P~m, let f denote the function of distance from Q to the points of Li. We show that f is convex. Let M be the projection point of Q on Li (or its extension), i.e., M is selected such that segment Q M is perpendicular to Li, and let d = d(Q, M) be the distance between Q and M. Take any point A E Li, and denote d(A,M) = x. Then since d is constant, d(Q,A) = f(x) = v / ~ + d 2, and so f " ( X ) --
d2 ( x2 + d2)3/2 _> 0,
which shows that f is convex on Li. Now r((Pm; Q), u) is defined for all u E S 1, and since a convex polygon is composed of a finite number of line segments over which f is convex for each segment, it is clear that r((Pm; Q), u) is a piecewise convex function. Note that since Q E p o , r((Pm; Q), u) > 0, for all u E S 1 and r((P,~; Q), u) is pieeewise convex, it follows that r((Pm; Q), u) -1 is a piecewise concave function. I An example is used to illustrate the numerical solution of problem (4). EXAMPLE. P6 = { ( - 7 , - 8 ) , ( - 4 , 3 ) , ( 5 , 7 ) , ( 5 . 5 , 3 ) , ( 5 , 1 ) , ( 3 , - 2 ) } .
The radial functional r((P6; Q),u) is constructed directly from P6 using the definition. From points Q = (-4.75,-4.33) and Q' = (0.25,0.67) as depicted in Figure 1, the piecewise-convex form of r((P6; Q), u) is clear. The radial function is then inverted and integrated for points selected over a grid throughout the domain P6. The resulting functional is shown in Figure 2 and the solution point for this example is located at (-0.23, 0.23). I This example suggests the following conjecture. CONJECTURE. f(x, y) = f r((Pm;Q),u) du is a convex functional and g(x, y) = f r((Pm; Q), u) du is a concave functional.
The continuous inverse Weber model (4) has an intriguing relationship to a geometric optimization problem that arises in convex geometry. PROPOSITION 2.1. For fixed Q E P°m, f r((Pm;Q),u) du is equal to the length (perimeter) of the polar polygon (Pro; Q). PROOF. Consider a general convex figure K, and recall that the length of K, L(K), is given by [3, p. 31
L(K) = J H ((K; x), u) du,
(5)
where H((K; Q), u) is the support function of K, H((K; Q), u) = sup{(u, x) I x E K}, and (u, x) is the inner product of u and x. Using Euclidean duality [4, pp. 60-64], the gauge function of K, d((K; Q), u) is equal to the support function of the polar figure (K*;Q), H((K*;Q), u), where the gauge function is related to the radial function by d((K; Q), u) = 1/r((K; Q), u) and the polar figure is defined by (K*; x) = {z' E R 2 [ (x' - x, y - x) _< 1, for all y E K}. Then from (5),
L(K*;Q) =
H ( ( K * ; Q ) , u ) du =
d((K;Q),u) du =
r((K;Q),u)"
(6) I
The length of the polar figure (K*; Q) is a function of the position of Q E K °, and the position of the minimum perimeter polar figure (K*; Q) is precisely the solution point to formulation (4) when K = Pro: du inf L ( P * ; Q ) = inf [ QaP& QeP& J r ((Pro; Q), u)" (7) /*
124
M. J. KAISER
X= -'t. 75{0[4{} ?=-'t,3333:]
14.88,
/
11.90.
\
# / 3'
.92~60.
\ \
\
y
5. 95240,
\ \
2. 97620,
0. 00000
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0
0
~;
,.:,
,~
0
0
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.A, '
0
0
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0
'
~ 0
. . . .
0
0
0
0
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0
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,~
0
0
0
. . . .
0
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X= !,-J.2~0~8 ~ = 0 . 6 6 5 6 7
l.t. 2 6 .
A
9 • 00990.
$A
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/ j'
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/ x..
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,,>
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o-.~.
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4.5049~
2,2324?
0. 00000 0
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0
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0
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0
Figure 1. The radial functional r(P6,u) at points Q = ( - 4 . 7 5 , - 4 . 3 3 ) (0.25, 0.67) for P6 = { ( - 7 , - 8 ) , ( - 4 , 3), (5, 7), (5.5, 3), (5, 1), (3, - 2 ) } .
0
0
0
0
and Q~ =
This provides an interesting geometric interpretation of the inverse Weber model. The LHS of (7) can be solved directly by computing the perimeter of the polar polygon L(Pm;Q) and then applying a simple exhaustive search to minimize this value [5]; a closed-form solution methodology expressing the length of (P*; Q) in terms of the reference point Q = (x, y) using symbolic techniques and solving via Newton's method is also possible [6]. Yet another solution procedure
Weber Location Models
Figure 2. The flmctional f
du
125
for P6.
elucidated here in the context of the 1-D inverse Weber problem and as demonstrated in the preceding example is to directly construct the radial functional r((P,~; Q),,u), compute the relevant integral, and then solve the optimization problem through an exhaustive search. Each of these methods has its advantages and limitations, e.g., the perimeter of the polar polygon can be computed directly from the input data but is very problem specific and the solution is still via a search procedure; the symbolic approach is beautiful and can be implemented quickly but it is tedious and its complexity grows with the size of Pm despite the quadratic convergence of the procedure; the direct geometric approach, which as elucidated here uses the radial function, requires a large initial programming investment but is appealing since it has broad and substantially more general applications. Considering the different solution methodologies which have been presented for fbrnmlation (7), and using analogy as a guide, it appears reasonable that a symbolic approach to the integral problem described by (4) and stated in the conjecture can also be developed. XVith this in mind, an alternative expression for the inverse Weber functional can be written as follows. Following Proposition 2.1, the integral functional can be expressed as follows. PROPOSITION 2.2 " f r((Pm;Q),u) " du =L(P~;(x,y)) fi =
m
=}-~i=lfi(x,Y),
where
x / ( a i + l b i _ bi+lai)2(x2 + y2) + 2(ai+lbi - bi+lai)[(ci+lbi - bi+ lCi)X + (ai+lci - c.~+lai)g] + ~ ( a i + l x + bi+ly 4- Ci+l)(a~x 4- biy ÷ ci)
where c~ = ai+lC~ -- C i + l a i + b~+lCi - c i + l b i a n d P'm = { L 1 , . . . , L , ~ } , Li : a,:r + b~y + c, = 0, i = l,...,m.
PROOF. From point Q E p o , the dual polygon (Pro; Q) is defined through the following algebraic correspondence. Input: P m = { A 1 , A 2 , . . . , A m } ; A i = (x~,yi), i = 1 , . . . , m ; Q = (x,y). Output: ( P * ; Q ) = { A ~ , A ~ , . . . , A ~ } ; A• = ( x * , y ; ) , i = 1 . . . . . m . The input data are the vertices A~ = (xi, Yi), i = 1 , . . . , m of the convex polygon P~ and the point Q = (x, y). The procedure to construct (P*; Q) is a two-step process. Step 1. Determine P'm = { L 1 , . . . , L m } , where L~ : a~x + b~y + ci = 0, i = 1 , . . . , m . Step 2. Determine A~' = ( a i / ( a i x + b i y + ci) + x , b ~ / ( a i x + b i y + ci) + y ) , i -- 1 , . . . .
m.
126
M . J . KAISER
The edges of Pm are computed in Step 1 and the polar correspondence is determined in Step 2 (after the duality point Q = (x, y) is selected) by simply reading off the coefficients of the lines Li and using the value of Q. This correspondence can be derived algebraically, or the correspondence can be accepted as the definition of polarity [7, pp. 46-51]. The polar polygon (P*; Q) is the convex hull of the "polar" vertices A i = (xi, Yi ), i = 1 , . . . , m, and the perimeter of (Pro, Q) is the sum of the length of the boundary segments; e.g., m
L (Pro, $° Q)
~- E i=l
d (A~, A*+I),
where A*+I = A~, for the polar vertices given in Step 2. Set fi = d(A~, Ai*+l ). Then,
fi ~
ai+l x + bi+lY +
Ci+l
) 2 (+ ai aix + ~ y + ci
bi+l ai+lx + bi+lY + ci+l
and expansion of this term yields the desired form.
b i )
2
aix + ~ y + ci
|
REFERENCES 1. R. Francis, L.F. McGinnis and J. White, Facility Layout and Location: An Analytical Approach, Second edition, Prentice Hall, Englewood Cliffs, N J, (1992). 2. G.B. Thomas and R.L. Finney, Calculus and Analytic Geometry, Seventh edition, Addison-Wesley, Reading, MA, (1988). 3. L.A. Santal6, Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA, (1976). 4. H.M. Guggenheimer, Applicable Geometry, Global and Local Convexity, R.E. Krieger, Huntington, NY, (1977). 5. M.J. Kaiser, The Blaschke-Steinhardt point of a planar convex set, Computers Math. Applic. 28 (7), 1-11, (1994). 6. M.J. Kaiser, The mixed volume optimization problem (manuscript), (1995). 7. B. Griinbaum, Convex Polytopes, Interscience Publishers, London, (1967).
Pergamon
Pll: S0893-9659(97)00144-4
Some Thoughts on Generalized Weber Location Models M. J. KAISER Department of Industrial and Manufacturing Engineering Wichita State University, Wichita, KS 67260-0035, U.S.A. kaiser©ie, twsu. edu
(Received January 1996; revised and accepted November 1996) Abstract--The Weber and inverse Weber location problem is defined for a continuous onedimensional convex region in the plane and solved using constructive numerical techniques. It is conjectured t h a t the Weber functional for a continuous one-dimensional convex region is concave. The equivalence between the one-dimensionai inverse Weber model and a polar geometric optimization problem is demonstrated, and an alternative symbolic expression for the integral functional is described. Keywords--Constructive geometry, Generalized Weber problem, Geometric analysis, Continuous location models, Polar optimization.
The single-source discrete (zero-dimensional) Weber problem is the prototype location model. Given a set of points S m = {A1,... , A m } , where Ai = (xi,yi), i = 1 , . . . ,m, that form the set of cities, find the location of the facility to minimize the sum of the Euclidean distances from (x, y) to Ai = (xi, Yi), d((x, y), Ai), where
d ((x, y), Ai) = x / ( x - xi) 2 + (y - yi) 2. The zero-dimensional Weber location problem can thus be written as m
min ~
rid ((x, y), A i ) ,
(1)
i=1
where ti is a positive number (weight) associated with Ai, i = 1 , . . . ,m. The function defined by (1) is convex, and numerical methods (such as Weiszfeld's Algorithm or Newton's Method) to solve for this functional-type are well known [1, Chapter 4]. A zero-dimensional "inverse" Weber location model can be formulated as
min~
ti d ((x, y), Ai)"
(2)
i=1
This model admits a natural interpretation in terms of locating noxious facilities, but the form of the functional in this case is not convex, and thus, the solution point(s) are not readily characterized. Typeset by A2~S-TEX 121
122
M.J. KAISER
The zero-dimensional (0-D) "point set" location models motivate the following generalization. Consider a one-dimensional (l-D) "continuous" region defined as the boundary of a convex figure K (a "convex curve"). A convex polygon Pm with interior p o and boundary OPm will be used as the realization of K. Pm can be described in terms of its vertices as Pm = {A1,..., Am}, Ai -- (xi, Yi), i = 1 , . . . ,m, or in terms of the line segments L~ that compose its boundary as P~m = { L 1 , . . . ,Lm}, Li : aix+biy+ci = 0, i = 1 , . . . ,rn. Nonweighted Weber and inverse Weber location models for a continuous 1-D region defined from OPm can thus be written as sup
QeP&
fo
r ((Pro; Q), u) du,
du inf fo 2~ QeP& r((Pm;Q),u)'
(3)
(4)
where r((Pm; Q), u) is the radial function defined for u E S 1, the unit circle by
r((Pm;Q),u) = max{A _> 0 I Ax • Pm, u • $1}. The radial function is what we typically think of in terms of polar coordinates as '%racing out" the boundary of the figure. The radial function is a directed distance with respect to the point Q • p o and direction defined by u • S 1, and therefore, r((Pm; Q), u) is a function of Q and u • S 1. The radial function describes the radial distance from point Q to the boundary OPrn as a function of the counterclockwise angle u from the horizontal x-axis. The ray from Q • P m° in )
the direction defined by u, Q(u) intersects OP'm at point T(u). The radial function is simply the distance from Q to T(u), i.e., r((Pm;Q),u) = d(Q,T(u)), as u revolves about S t. For a small class of symmetric regions, the radial functional can be described in closed parametric form at the point of symmetry; in general, however, no simple representation exists for the radial function defined at any arbitrary point within a convex polygon. In general, r((Pm; Q), u) is a complicated functional dependent on the geometry of Pm which can only be solved numerically. As Q approaches the boundary of Pm, it is clear that r((Pm; Q), u) will approach zero, and the functional defined by (4) will subsequently "blow up". This implies that the natural formulation of (4) should be stated as a minimization problem. Model (3) represents (after the inclusion of a 1/2~r-term) the average radial value to the boundary OPm with respect to the reference point Q • Pr°n, and it is interesting that there is an extremal formulation associated with this problem. One possible interpretation of model (3) is as follows. Imagine defending a region where hostile forces frequently attack, and you want to locate your castle to maximize its average distance away from the boundary. The natural objective function to consider is described by (3) since a priori knowledge of the attack front is not known. An interpretation of model (4) is provided in Proposition 2.1. A "naive" solution methodology suggested by formulations (1),(2) does not work for the continuous Weber models. T h a t is, by selecting successively larger sets of points Ai • OPm, and then solving (1) or (2) "in the limit" would yield an unbounded functional value and no distinction between arbitrary interior points Q and Q' within Pro. The integration procedure is the key to avoid this infinity in the 1-D case and is a natural generalization to consider since it leads to "average" functional values. Recall [2, p. 620, Ex. 30]: the average value of r over the curve r = f(O), c~ < 0
f(O) dO.
In the case of a circle of radius a centered at the origin O, a = 0, f~ = 2~r, (r(f(O); 0)) = a. As the reference point Q moves within the curve, however, the value of the integral will change in a complicated fashion.
Weber Location Models
123
By analogy with the Euclidean distance function d((x, y), As), it is shown that for a convex polygon and fixed point Q E P°m, the radial functional r((Pm; Q), u) is a piecewise convex function of u E S 1. Since the distance function is convex, it is clear that the radial functional will be piecewise convex. This can also be shown as follows. PROPOSITION 1. For fixed Q e po, r( (Pm; Q), u) is a piecewise convex function. PROOF. Given point Q E F ° and the line segment Li of the polygonal representation P~m, let f denote the function of distance from Q to the points of Li. We show that f is convex. Let M be the projection point of Q on Li (or its extension), i.e., M is selected such that segment Q M is perpendicular to Li, and let d = d(Q, M) be the distance between Q and M. Take any point A E Li, and denote d(A,M) = x. Then since d is constant, d(Q,A) = f(x) = v / ~ + d 2, and so f " ( X ) --
d2 ( x2 + d2)3/2 _> 0,
which shows that f is convex on Li. Now r((Pm; Q), u) is defined for all u E S 1, and since a convex polygon is composed of a finite number of line segments over which f is convex for each segment, it is clear that r((Pm; Q), u) is a piecewise convex function. Note that since Q E p o , r((Pm; Q), u) > 0, for all u E S 1 and r((P,~; Q), u) is pieeewise convex, it follows that r((Pm; Q), u) -1 is a piecewise concave function. I An example is used to illustrate the numerical solution of problem (4). EXAMPLE. P6 = { ( - 7 , - 8 ) , ( - 4 , 3 ) , ( 5 , 7 ) , ( 5 . 5 , 3 ) , ( 5 , 1 ) , ( 3 , - 2 ) } .
The radial functional r((P6; Q),u) is constructed directly from P6 using the definition. From points Q = (-4.75,-4.33) and Q' = (0.25,0.67) as depicted in Figure 1, the piecewise-convex form of r((P6; Q), u) is clear. The radial function is then inverted and integrated for points selected over a grid throughout the domain P6. The resulting functional is shown in Figure 2 and the solution point for this example is located at (-0.23, 0.23). I This example suggests the following conjecture. CONJECTURE. f(x, y) = f r((Pm;Q),u) du is a convex functional and g(x, y) = f r((Pm; Q), u) du is a concave functional.
The continuous inverse Weber model (4) has an intriguing relationship to a geometric optimization problem that arises in convex geometry. PROPOSITION 2.1. For fixed Q E P°m, f r((Pm;Q),u) du is equal to the length (perimeter) of the polar polygon (Pro; Q). PROOF. Consider a general convex figure K, and recall that the length of K, L(K), is given by [3, p. 31
L(K) = J H ((K; x), u) du,
(5)
where H((K; Q), u) is the support function of K, H((K; Q), u) = sup{(u, x) I x E K}, and (u, x) is the inner product of u and x. Using Euclidean duality [4, pp. 60-64], the gauge function of K, d((K; Q), u) is equal to the support function of the polar figure (K*;Q), H((K*;Q), u), where the gauge function is related to the radial function by d((K; Q), u) = 1/r((K; Q), u) and the polar figure is defined by (K*; x) = {z' E R 2 [ (x' - x, y - x) _< 1, for all y E K}. Then from (5),
L(K*;Q) =
H ( ( K * ; Q ) , u ) du =
d((K;Q),u) du =
r((K;Q),u)"
(6) I
The length of the polar figure (K*; Q) is a function of the position of Q E K °, and the position of the minimum perimeter polar figure (K*; Q) is precisely the solution point to formulation (4) when K = Pro: du inf L ( P * ; Q ) = inf [ QaP& QeP& J r ((Pro; Q), u)" (7) /*
124
M. J. KAISER
X= -'t. 75{0[4{} ?=-'t,3333:]
14.88,
/
11.90.
\
# / 3'
.92~60.
\ \
\
y
5. 95240,
\ \
2. 97620,
0. 00000
'
0
0
~;
,.:,
,~
0
0
C)
.A, '
0
0
0
0
'
~ 0
. . . .
0
0
0
0
0
0
0
0
/ ,; 0
'
,~
0
0
0
. . . .
0
0
0
0
0
0
0
0
X= !,-J.2~0~8 ~ = 0 . 6 6 5 6 7
l.t. 2 6 .
A
9 • 00990.
$A
6. 7374~-,
/ j'
/ x....
/ x..
/
,,>
/
o-.~.
"x..
4.5049~
2,2324?
0. 00000 0
0
0
0
0
0
0
~
0
0
0
0
0
0
Figure 1. The radial functional r(P6,u) at points Q = ( - 4 . 7 5 , - 4 . 3 3 ) (0.25, 0.67) for P6 = { ( - 7 , - 8 ) , ( - 4 , 3), (5, 7), (5.5, 3), (5, 1), (3, - 2 ) } .
0
0
0
0
and Q~ =
This provides an interesting geometric interpretation of the inverse Weber model. The LHS of (7) can be solved directly by computing the perimeter of the polar polygon L(Pm;Q) and then applying a simple exhaustive search to minimize this value [5]; a closed-form solution methodology expressing the length of (P*; Q) in terms of the reference point Q = (x, y) using symbolic techniques and solving via Newton's method is also possible [6]. Yet another solution procedure
Weber Location Models
Figure 2. The flmctional f
du
125
for P6.
elucidated here in the context of the 1-D inverse Weber problem and as demonstrated in the preceding example is to directly construct the radial functional r((P,~; Q),,u), compute the relevant integral, and then solve the optimization problem through an exhaustive search. Each of these methods has its advantages and limitations, e.g., the perimeter of the polar polygon can be computed directly from the input data but is very problem specific and the solution is still via a search procedure; the symbolic approach is beautiful and can be implemented quickly but it is tedious and its complexity grows with the size of Pm despite the quadratic convergence of the procedure; the direct geometric approach, which as elucidated here uses the radial function, requires a large initial programming investment but is appealing since it has broad and substantially more general applications. Considering the different solution methodologies which have been presented for fbrnmlation (7), and using analogy as a guide, it appears reasonable that a symbolic approach to the integral problem described by (4) and stated in the conjecture can also be developed. XVith this in mind, an alternative expression for the inverse Weber functional can be written as follows. Following Proposition 2.1, the integral functional can be expressed as follows. PROPOSITION 2.2 " f r((Pm;Q),u) " du =L(P~;(x,y)) fi =
m
=}-~i=lfi(x,Y),
where
x / ( a i + l b i _ bi+lai)2(x2 + y2) + 2(ai+lbi - bi+lai)[(ci+lbi - bi+ lCi)X + (ai+lci - c.~+lai)g] + ~ ( a i + l x + bi+ly 4- Ci+l)(a~x 4- biy ÷ ci)
where c~ = ai+lC~ -- C i + l a i + b~+lCi - c i + l b i a n d P'm = { L 1 , . . . , L , ~ } , Li : a,:r + b~y + c, = 0, i = l,...,m.
PROOF. From point Q E p o , the dual polygon (Pro; Q) is defined through the following algebraic correspondence. Input: P m = { A 1 , A 2 , . . . , A m } ; A i = (x~,yi), i = 1 , . . . , m ; Q = (x,y). Output: ( P * ; Q ) = { A ~ , A ~ , . . . , A ~ } ; A• = ( x * , y ; ) , i = 1 . . . . . m . The input data are the vertices A~ = (xi, Yi), i = 1 , . . . , m of the convex polygon P~ and the point Q = (x, y). The procedure to construct (P*; Q) is a two-step process. Step 1. Determine P'm = { L 1 , . . . , L m } , where L~ : a~x + b~y + ci = 0, i = 1 , . . . , m . Step 2. Determine A~' = ( a i / ( a i x + b i y + ci) + x , b ~ / ( a i x + b i y + ci) + y ) , i -- 1 , . . . .
m.
126
M . J . KAISER
The edges of Pm are computed in Step 1 and the polar correspondence is determined in Step 2 (after the duality point Q = (x, y) is selected) by simply reading off the coefficients of the lines Li and using the value of Q. This correspondence can be derived algebraically, or the correspondence can be accepted as the definition of polarity [7, pp. 46-51]. The polar polygon (P*; Q) is the convex hull of the "polar" vertices A i = (xi, Yi ), i = 1 , . . . , m, and the perimeter of (Pro, Q) is the sum of the length of the boundary segments; e.g., m
L (Pro, $° Q)
~- E i=l
d (A~, A*+I),
where A*+I = A~, for the polar vertices given in Step 2. Set fi = d(A~, Ai*+l ). Then,
fi ~
ai+l x + bi+lY +
Ci+l
) 2 (+ ai aix + ~ y + ci
bi+l ai+lx + bi+lY + ci+l
and expansion of this term yields the desired form.
b i )
2
aix + ~ y + ci
|
REFERENCES 1. R. Francis, L.F. McGinnis and J. White, Facility Layout and Location: An Analytical Approach, Second edition, Prentice Hall, Englewood Cliffs, N J, (1992). 2. G.B. Thomas and R.L. Finney, Calculus and Analytic Geometry, Seventh edition, Addison-Wesley, Reading, MA, (1988). 3. L.A. Santal6, Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA, (1976). 4. H.M. Guggenheimer, Applicable Geometry, Global and Local Convexity, R.E. Krieger, Huntington, NY, (1977). 5. M.J. Kaiser, The Blaschke-Steinhardt point of a planar convex set, Computers Math. Applic. 28 (7), 1-11, (1994). 6. M.J. Kaiser, The mixed volume optimization problem (manuscript), (1995). 7. B. Griinbaum, Convex Polytopes, Interscience Publishers, London, (1967).