Location of a line in the three-dimensional space Daniel Scholz - - PowerPoint PPT Presentation

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results

Location of a line in the three-dimensional space

Daniel Scholz

Institute for Numerical and Applied Mathematics, University of G¨

• ttingen

May 27, 2010

Doc-Course: Constructive Approximation, Optimization and Mathematical Modeling, Sevilla, Spain

Supervised by

Rafael Blanquero, Emilio Carrizosa, and Anita Sch¨

• bel

Daniel Scholz University of G¨

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results

Outline

• 1. Introduction

1.1 The Weber problem 1.2 Line location in 2D 1.3 Line location in 3D

• 2. Problem formulation

2.1 Closed formula 2.2 Properties 2.3 Parametrization

• 3. Geometric branch-and-bound

3.1 The algorithm 3.2 Initial box 3.3 Lower bounds

• 4. Numerical results

4.1 Example instance 4.2 Random input data 4.3 Discussion

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Facility locations problems Figure: Some given demand points on the plane.

Weber problem: Minimize the sum of the distances between a new facility and the demand points.

Daniel Scholz University of G¨

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May 27, 2010 3 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Facility locations problems Figure: Some given demand points on the plane.

Weber problem: Minimize the sum of the distances between a new facility and the demand points.

Daniel Scholz University of G¨

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May 27, 2010 3 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Line location in the two-dimensional space Figure: Some given demand points on the plane.

Median line problem: All optimal solutions pass through two demand points, see Korneenko and Martini (1993) or Sch¨

• bel (1999).

Daniel Scholz University of G¨

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May 27, 2010 4 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Line location in the two-dimensional space Figure: Some given demand points on the plane.

Median line problem: All optimal solutions pass through two demand points, see Korneenko and Martini (1993) or Sch¨

• bel (1999).

Daniel Scholz University of G¨

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May 27, 2010 4 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Line location in the three-dimensional space

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Figure: Some given demand points in the three-dimensional space.

Objective: Find a line which minimizes the sum of the distances between such line and the given demand points.

Daniel Scholz University of G¨

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May 27, 2010 5 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Line location in the three-dimensional space

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Figure: Some given demand points in the three-dimensional space.

Objective: Find a line which minimizes the sum of the distances between such line and the given demand points.

Daniel Scholz University of G¨

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May 27, 2010 5 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The Weber problem :Line location in two dimensions :Line location in three dimensions

Known results for the three-dimensional median line problem

Some special cases are discussed in Brimberg, Juel, Sch¨

• bel (2002) and in

Brimberg, Juel, Sch¨

• bel (2003):
• 1. Locating a vertical line can be reduced to a Weber problem in the

two-dimensional space.

• 2. The case that all demand points are contained in a hyperplane yields

basically a two-dimensional median line problem. However, there is no solution method for the general median line problem in the three-dimensional.

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May 27, 2010 6 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Closed formular for the three-dimensional median line problem (1)

A line r in R3 has the form r = r(x, d) = {x + td : t ∈ R} where d ∈ R3 \ {0} is the direction of r and x ∈ R3.

Notation

For any a ∈ R3 and x, d ∈ R3 with d = 0 denote by δ(r) = δa(x, d) = min

t∈R x + td − a2

the Euclidean distance from a to the line r = r(x, d). Hence, the median line problem can be formulated as follows: min

x,d∈R3 d=0

n

• k=1

δak(x, d).

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Closed formular for the three-dimensional median line problem (2)

Lemma 1

For any a ∈ R3 and x, d ∈ R3 with d = 0 we find δa(x, d) =

• x − a2

2 − (dT(a − x))2

dTd . Hence, the median line problem becomes min

x,d∈R3 d=0

n

• k=1
• x − ak2

2 − (dT(ak − x))2

dTd where A = {a1, . . . , an} ⊂ R3 are the given demand points.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Properties (1)

Obviously, the line r(x, d) is not uniquely defined by the pair (x, d). One finds r(x, d) = r(x + νd, d) for any ν ∈ R. Hence, we assume that x is the intersection of r with the hyperplane Hd = {y : dTy = 0}.

Corollary 2

For any a ∈ R3 and x, d ∈ R3 with d = 0 and dTx = 0 we find δa(x, d) =

• x − a2

2 − (dTa)2

dTd .

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Properties (2)

Moreover, we have r(x, d) = r(x, τd) for any τ = 0. Hence, it can be assumed that d = 1.

Corollary 3

For any a ∈ R3 and x, d ∈ R3 with d = 1 and dTx = 0 we find δa(x, d) =

• x − a2

2 − (dTa)2.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Properties (3)

Lemma 4

The (three-dimensional) median line problem with fixed direction d ∈ R3 \ {0} is equivalent to a (two-dimensional) Weber problem.

Figure: Solution method for the median line problem with fixed direction d.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Properties (4)

Corollary 5

It exists an optimal solution (x∗, d∗) ∈ R6 to the median line problem such that the line r = r(x∗, d∗) intersects the convex hull of A. This is a direct consequence of Lemma 4 since it is well-known that there exists an optimal solution for the Weber problem which intersects the convex hull of the (projected) demand points, see Wesolowsky (1975).

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Problem parametrization (1)

If we consider only d = (d1, d2, d3) ∈ R3 such that d3 = 1 as well as x = (x1, x2, x3) ∈ R3 such that dTx = 0, we obtain x3 = − (x1d1 + x2d2). Hence, Corollary 2 yields f1(x1, x2, d1, d2) =

n

• k=1
• gk

1 (x1, x2, d1, d2)

and with ak = (αk, βk, γk) for k = 1, . . . , n we have

g k

1 (x1, x2, d1, d2) = (x1−αk)2+(x2−βk)2+(x1d1+x2d2+γk)2−(d1αk + d2βk + γk)2

d2

1 + d2 2 + 1

.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Closed formula :Properties :Problem parametrization

Problem parametrization (2)

In the same way we can also fix d1 = 1 and d2 = 1 which yields f2(x1, x2, d1, d2) =

n

• k=1
• gk

2 (x1, x2, d1, d2),

f3(x1, x2, d1, d2) =

n

• k=1
• gk

3 (x1, x2, d1, d2).

To sum up, the six-dimensional median line problem is equivalent to the four-dimension problem min

x1,x2,d1,d2∈R i∈{1,2,3}

fi(x1, x2, d1, d2).

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (1)

Assume an objective function f : Rn → R and a feasible area X ⊂ Rn which is a box, i.e. a Cartesian product of intervals. Goal: Minimize f on X. To this end, we assume that some lower bounds for all subboxes Y = [x1, x1] × . . . × [xn, xn] ⊂ X can be calculated, i.e. we assume that we are in a possition to find values LB(Y ) ∈ R such that LB(Y ) ≤ f (x) for all x ∈ Y .

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

Daniel Scholz University of G¨

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May 27, 2010 16 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

Daniel Scholz University of G¨

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May 27, 2010 16 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

Daniel Scholz University of G¨

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May 27, 2010 16 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

Daniel Scholz University of G¨

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May 27, 2010 16 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

Daniel Scholz University of G¨

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May 27, 2010 16 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (3) Figure: The idea of geometric branch-and-bound methods.

Daniel Scholz University of G¨

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May 27, 2010 16 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The general idea of geometric branch-and-bound methods (2)

Goal: Minimize f : Rn → R on a box X ⊂ Rn.

• 1. Calculate a lower bound LB(X) and set UB = f (c(X)).
• 2. Choose a box with the greatest diameter, split it into s congruent

smaller boxes Y1, . . . , Ys, and delete the selected box. Calculate lower bounds LB(Y1), . . . , LB(Ys) and update UB = min{UB, f (c(Y1)), . . . , f (c(Ys))}. Delete all boxes Y with LB(Y ) + ε ≥ UB.

• 3. When there are no boxes left, the algorithm terminates and UB is

within an accuracy of ε from the optimum. If there are boxes left, return to step 2.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Remarks on the algorithm

Of course, the most important and the most difficult task throughout the algorithm is to calculate good lower bounds. Geometric branch-and-bound in location theory: Hansen et al. (1985), Plastria (1992), Drezner and Suzuki (2004), Drezner (2007), Blanquero and Carrizosa (2009), Sch¨

• bel and S. (2010).

Applications in location theory: Drezner (2007), Sch¨

• bel and S. (2010),

Blanquero et al. (2009) among plenty of other references.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Lower bounds obtained from d.c. decompositions (1) Figure: Lower bounds for d.c. functions.

Let g : Rn → R be convex and ξ a subgradient of g at c ∈ Rn. Then a(x) := g(c) + ξT(x − c) ≤ g(x) for all x ∈ Rn and a is a linear function.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Lower bounds obtained from d.c. decompositions (1) Figure: Lower bounds for d.c. functions.

Let g : Rn → R be convex and ξ a subgradient of g at c ∈ Rn. Then a(x) := g(c) + ξT(x − c) ≤ g(x) for all x ∈ Rn and a is a linear function.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Lower bounds obtained from d.c. decompositions (2)

Let f (x) = g(x) − h(x) be a d.c. decomposition of f , i.e. g and h are convex functions. Then m(x) := a(x) − h(x) = g(c) + ξT(x − c) − h(x) ≤ f (x) for all x ∈ Rn and m : Rn → R is a concave function. Hence, for any box Y ⊂ Rn LB(Y ) = min{m(v1), . . . , m(v2n)} ≤ f (x) for all x ∈ Rn. Here, v1, . . . , v2n are the 2n vertices of Y .

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Lower bounds obtained from interval analysis

Interval analysis is a general framework for calculations with intervals, see e.g. Hansen (1992) or Ratschek and Rokne (1988). If F is the natural interval extension of f : Rn → R, we obtain f (Y ) ⊂ F(Y ) for all boxes Y ⊂ Rn. Hence, we find the lower bound LB(Y ) = F(Y )L, see Hansen (1992).

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Lower bounds for the median line problem (1)

Recall that for any subbox Y = X1 × X2 × D1 × D2 ⊂ R4 we want to find a lower bound on the median line objective functions fi(x1, x2, d1, d2) =

n

• k=1
• gk

i (x1, x2, d1, d2)

for i = 1, 2, 3. To this end, assume that some concave functions hk

i : R4 → R are known

for i = 1, 2, 3 and k = 1, . . . , n such that 0 ≤ hk

i (x1, x2, d1, d2) ≤ gk i (x1, x2, d1, d2)

for all (x1, x2, d1, d2) ∈ Y .

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

Lower bounds for the median line problem (2)

Theorem 6

Define hi(x1, x2, d1, d2) =

n

• k=1
• hk

i (x1, x2, d1, d2)

for i = 1, 2, 3. Then hi is a concave function and hi(x1, x2, d1, d2) ≤ fi(x1, x2, d1, d2) for all (x1, x2, d1, d2) ∈ Y and i = 1, 2, 3. This result easily leads to lower bounds for the median line problem.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :The prototype algorithm :Lower bounds :Initial box

The initial box - a dominating location set

Theorem 7

Assume that {a1, . . . , an} ⊂ [−1, 1]3. Then the initial box X = [− √ 3, √ 3] × [− √ 3, √ 3] × [−1, 1] × [−1, 1] contains at least one optimal solution to the median line problem.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

A particular example instance (1) With ε = 10−6, we obtain after 976 861 iterations the solution r = r(x∗, d∗) =   1.021705 1.173660 1.119308   + t ·   −0.980400 1.000000 −0.153648   with an objective value of 36.893231.

(1.6, 0.2, 0.0) (0.5, 0.4, 1.0) (0.3, 1.8, 1.8) (0.7, 1.4, 1.5) (1.5, 1.8, 0.7) (0.8, 2.0, 1.2) (2.0, 1.8, 0.0) (1.3, 0.6, 0.5) (1.7, 0.1, 1.6) (0.4, 1.4, 0.2) (1.4, 1.2, 0.1) (1.7, 0.3, 1.2) (0.7, 2.0, 1.1) (0.8, 1.2, 0.8) (1.6, 1.7, 0.8) (0.1, 1.5, 0.2) (1.9, 0.6, 1.6) (1.9, 0.9, 1.0) (2.0, 0.2, 0.1) (2.0, 0.6, 1.2) (0.0, 0.4, 0.8) (1.6, 1.0, 0.8) (0.7, 1.0, 2.0) (1.7, 0.1, 1.9) (0.3, 1.5, 1.1) (1.0, 1.9, 1.4) (0.5, 1.5, 0.9) (0.4, 0.7, 1.1) (0.8, 0.9, 2.0) (1.9, 0.2, 1.6) (0.8, 1.3, 1.4) (1.8, 1.8, 0.6) (1.5, 1.1, 1.6) (0.3, 0.9, 2.0) (0.8, 0.1, 2.0) (0.8, 1.1, 0.3) (2.0, 1.8, 1.6) (1.6, 1.5, 0.8) (0.2, 2.0, 1.2) (1.2, 1.6, 0.7) (1.8, 1.4, 1.8) (0.1, 1.2, 1.1) (1.1, 0.3, 0.6) (1.9, 1.4, 0.3) (0.0, 0.9, 0.1) (0.7, 1.5, 1.1) (1.5, 1.2, 1.6) (1.6, 0.0, 1.3) (1.3, 1.7, 1.3) (0.5, 0.0, 0.3)

Table: Input data A = {a1, . . . , a50}.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

A particular example instance (2)

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Figure: Optimal solution for a particular problem instance.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

Results for randomly generated input data

For various n we solved 10 problem instances with randomly input data.

Run time (sec.) Iterations n Min Max Ave. Min Max Ave. 5 0.53 32.60 4.41 86, 784 4, 815, 863 652, 041.4 10 1.95 99.58 40.60 147, 241 7, 467, 943 3, 115, 063.1 15 3.78 110.17 28.44 191, 174 5, 516, 327 1, 452, 053.0 20 4.57 90.59 40.55 175, 005 3, 395, 947 1, 569, 748.8 25 6.99 57.14 29.67 213, 679 1, 786, 429 925, 439.1 30 7.78 45.38 29.25 205, 987 1, 187, 215 769, 554.0 35 21.65 144.64 67.04 491, 107 3, 219, 516 1, 503, 225.0 40 26.55 104.01 56.78 529, 150 2, 105, 972 1, 133, 395.8 45 21.42 254.27 63.30 376, 893 4, 609, 846 1, 128, 499.6 50 17.50 148.19 63.45 277, 817 2, 362, 796 1, 006, 704.5 Table: Numerical results with randomly generated ak ∈ [−1, 1]3 and ε = 10−6.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

Summary and discussion Summary

• 1. We studied the median line problem in the three-dimensional

Euclidean space.

• 2. As solution algorithm, we suggested a geometric branch-and-bound

method and some lower bounds were presented.

• 3. Numerical results showed that problem instances with up to n = 50

demand points can be solved in a few minutes of computer time.

Discussion

• 1. Note that of course other parametrizations and bounding
• perations for the median line problem are suitable.
• 2. An interesting further idea is to investigate more general distance

functions.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

Thank you!

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

References

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

References (1)

• R. Blanquero, E. Carrizosa (2009):

Continuous location problems and big triangle small triangle: Constructing better bounds. Journal of Global Optimization, 45: 389–402.

• R. Blanquero, E. Carrizosa, P. Hansen (2009):

Locating objects in the plane using global optimization techniques. Mathematics of Operations Research, 34: 837–858.

• J. Brimberg, H. Juel, A. Sch¨
• bel (2002):

Linear facility location in three dimensions - Models and solution methods. Operations Research, 50: 1050–1057.

• J. Brimberg, H. Juel, A. Sch¨
• bel (2003):

Properties of three-dimensional median line location models. Annals of Operations Research, 122: 71–85.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

References (2)

• Z. Drezner, A. Suzuki (2004):

The big triangle small triangle method for the solution of nonconvex facility location problems. Operations Research, 52: 128–135.

• Z. Drezner (2007):

A general global optimization approach for solving location problems in the plane. Journal of Global Optimization, 37: 305–319.

• E. Hansen (1992):

Global Optimization Using Interval Analysis. Marcel Dekker, New York, 1st edition.

• P. Hansen, D. Peeters, D. Richard, J.F. Thisse (1985):

The minisum and minimax location problems revisited. Operations Research, 33: 1251–1265.

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:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

References (3)

• R. Horst, H. Tuy (1996):

Global Optimization: Deterministic Approaches. Springer, Berlin, 3rd edition. N.M. Korneenko, H. Martini (1993): Hyperplane approximation and related

• topics. In J. Pach:

New Trends in Discrete and Computational Geometry. Springer, New York: 135–162.

• F. Plastria (1992):

GBSSS: The generalized big square small square method for planar single-facility location. European Journal of Operational Research, 62: 163–174.

• H. Ratschek, J. Rokne (1988):

New Computer Methods for Global Optimization. Ellis Horwood, Chichester, England, 1st edition.

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May 27, 2010 33 / 34

:Introduction :Problem formulation :Geometric branch-and-bound :Numerical results :Example instance :Random input data :Discussion

References (4)

• A. Sch¨
• bel (1999):

Locating Lines and Hyperplanes. Theory and Algorithms. Kluwer Academic Publisher, Dordrecht, 1st edition.

• A. Sch¨
• bel, D. Scholz (2010):

The big cube small cube solution method for multidimensional facility location problems. Computers and Operations Research, 37: 115–122.

• H. Tuy (1998):

Convex Analysis, Global Optimization. Kluwer Academic Publisher, Dordrecht, 1st edition. G.O. Wesolowsky (1975): Location of the median line for weighted points. Environment and Planning A, 7: 163–170.

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