Locating a Semi-Obnoxious Facility with Repelling Polygonal Regions

Transcription

1 with Repelling Polygonal Regions, Universidad de Sevilla Frank Plastria, Vrije Universiteit Brussel Emilio Carrizosa, Universidad de Sevilla Baeza, 19th March 2007

2 Outline 1 Location of a semi-obnoxious facility Motivation of this problem 2 3 Multi-instance Classification Conclusions and future work

3 Location of a semi-obnoxious facility Motivation of this problem Examples of semi-obnoxious facilities

4 Location of a semi-obnoxious facility For the last years, the location of semi-desirable Motivation facilities of this hasproblem been a widely studied topic by the researchers in location theory (see [1, 2, 3, 4, 5, 6, 7, 8]). A Bibliography References [1] R. Blanquero and E. Carrizosa. A D.C. Biobjective Location Model. Journal of Global Optimization, 23: , (2002). [2] E. Carrizosa and F. Plastria. Location of Semi-Obnoxious Facilities. Studies in Locational Analysis, 12:1 27, (1999). [3] P. C. Chen, P. Hansen, B. Jaumard, and H. Tuy. Weber s Problem with Attraction and Repulsion. Journal of Regional Science, 32: , (1992). [4] E. Erkut and S. Neuman. Analytical Models for Locating Undesirable Facilities. European Journal of Operational Research, 40: , (1989). [5] S. Nickel and E. M. Dudenhoffer. Weber s Problem with Attraction and Repulsion under Polyhedral Gauges. Journal of Global Optimization, 11: , (1997). [6] Y. Ohsawa. Bicriteria Euclidean Location Associated with Maximin and Minimax Criteria. Naval Research Logistics, 47: , (2000). [7] Y. Ohsawa, F. Plastria, and K. Tamura. Euclidean Push-Pull Partial Covering Problems. Computers and Operations Research, 33: , (2006). [8] H. Tuy, F. Al-Khayal, and F. Zhou. A D.C. Optimization Method for Single Facility Location Problems. Journal of Global Optimization, 7: , (1995).

5 Location of a semi-obnoxious facility Motivation of this problem Semi-obnoxious facility in our problem Location of a semi-obnoxious facility in the euclidean plane. Attracting points: demand to be satisfied. Repelling regions: populated areas to be protected.

6 Initial situation Location of a semi-obnoxious facility Motivation of this problem

7 Our problem Location of a semi-obnoxious facility Motivation of this problem

8 Which ball? Location of a semi-obnoxious facility Motivation of this problem

9 Maximizing the margin Location of a semi-obnoxious facility Motivation of this problem

10 Multiple instance classification Location of a semi-obnoxious facility Motivation of this problem Circles: instances of G +1. Squares: instances of G 1. Bags: sets of instances (circles/squares) with the same colour.

11 Classification rule Location of a semi-obnoxious facility Motivation of this problem Multi-Instance Problem Problem of supervised classification where the objects to be classified are bags of instances measuring d attributes. A label +1 or -1 is assigned to each bag. Classification rule A bag is positive (label +1) if AT LEAST ONE of its instances satisfies a determined condition. A bag is negative (label -1) if NONE of the instances satisfies the condition.

12 Multiple instance classification Location of a semi-obnoxious facility Motivation of this problem Circles: instances of G +1. Squares: instances of G 1. Bags: sets of instances (circles/squares) with the same colour.

13 Our aim Dimension 2. Groups of customers G + = {x 1,..., x n } R 2, set of points (attracting elements). G = {S 1,..., S m } R 2, set of convex polygons (repelling elements). Construction of a ball B(x 0, r) such that d 2 (x 0, x i ) < r 2, x i G + d 2 (x 0, S j ) r 2, S j G

14 Our aim Dimension 2. Groups of customers G + = {x 1,..., x n } R 2, set of points (attracting elements). G = {S 1,..., S m } R 2, set of convex polygons (repelling elements). Construction of a ball B(x 0, r) such that d 2 (x 0, x i ) < r 2, x i G + d 2 (x 0, S j ) r 2, S j G

15 Margin strategy These constraints can be written as... r 2 x 0 x i 2 > 0, x i G + min x i G + (r 2 x 0 x i 2 ) > 0, min( x 0 x 2 r 2 ) 0, S j G min min( x 0 x 2 r 2 ) 0. x S j S j G x S j Margin = min { } min (r 2 x 0 x i 2 ), min min( x 0 x 2 r 2 ). x i G + S j G x S j

16 Margin strategy These constraints can be written as... r 2 x 0 x i 2 > 0, x i G + min x i G + (r 2 x 0 x i 2 ) > 0, min( x 0 x 2 r 2 ) 0, S j G min min( x 0 x 2 r 2 ) 0. x S j S j G x S j Margin = min { } min (r 2 x 0 x i 2 ), min min( x 0 x 2 r 2 ). x i G + S j G x S j

17 Margin strategy

18 Formulation Our problem remains as... max x 0,r, s.t. min (r 2 x 0 x i 2 ) x i G + min min( x 0 x 2 r 2 ) S j G x S j

19 Formulation If we denote by r 2 + = r 2 and r 2 = r 2 +, max x 0 R 2, r +,r 0 r 2 r+ 2 s.t. x 0 x i 2 r+, 2 x i G + x 0 x 2 r, 2 x S j, S j G.

20 Two concentric balls

21 Active elements A point x i from G + is an active point for the solution (x 0, r +, r ) iff d(x 0, x i ) = x 0 x i = r +. Set of active points: A + (x 0 ). A polygon S j from G is an active polygon for (x 0, r +, r ) iff d(x 0, S j ) = min x Sj x 0 x = r. Set of active polygons: A (x 0 ).

22 Active elements

23 Condition 1 If (x 0, r +, r ) is an optimal solution, then there exists at least one point in each set of active elements, A + and A.

24 Proof

25 Proof

26 Condition 2 Let (x 0, r +, r ) be an optimal solution, one has that: 1 If r + r, then there must exist at least two active elements in G. 2 If r + r, then there must exist at least two active points in G +.

27 Condition 2

28 For r + r... S y a r+ x0 r+' x0' r 2 r 2 + > r 2 r 2 +

29 and for r r +... S S a y z a y r- r-' r- r-' x0 x0' x0' x0 r 2 r 2 + > r 2 r 2 +

30 Condition 3 If (x 0, r +, r ) is an optimal solution, then the intersection of the convex hulls of the two groups of active elements A + and A is a non-empty set.

31 Condition 3

32 Condition 3

33 Consequently... If the intersection of the convex hulls of the two groups G + and G is an empty set, that is, CH(G + ) CH(G ) =, then the solution is unbounded and the separating balls are transformed into hyperplanes.

34 Consequence

35 Consequence

36 Bisector of two polygons 1) 2) 3) 1 Mediatrix (two points) 2 Parabola (one point and one edge) 3 Bisectrix of the angle (two edges)

37 Bisector of two polygons S1 S2

38 Data in generic position FORBIDDEN CASES: 1 One point of G + and two vertices of two polygons of G are collinear. 2 One point of G + and one vertex of a polygon of G define an orthogonal direction to an edge of a polygon of G. 3 Two points of G + and one vertex of a polygon of G are collinear. 4 Two points of G + define an orthogonal direction to an edge of a polygon of G.

39 Different cases (for r + > r ) a1 a1 S S y y x0 y0 a0 x0 a0 y0 a2 a2

40 Different cases (for r + < r ) Z a y1 S1 x0 y2 S2

41 Different cases (for r + < r ) S1 y1 Z a x0 y2 S2

42 Different cases (for r + < r ) y1 z S1 a x0 x0' y0 z0 a0 y2 S2

43 Condition 4 Under the assumption that the data are in general position, if (x 0, r +, r ) is an optimal solution, one of the following situations arises: 1 there exist at least four active elements associated; 2 there exist three active elements, two polygons S 1, S 2 A and one point a A +, satisfying that y 1, a and y 2 are collinear, with y i such that d(x 0, S i ) = min x Si d(x 0, S i ) = d(x 0, y i ), i = 1, 2; 3 there exist three active elements, two polygons S 1, S 2 A and one point a A +, and x 0 is a breakpoint.

44 Nearest and farthest-point Voronoi diagrams Given the set of points {x 1,..., x n } and the set of polygons {S 1,..., S m }, Farthest-point Voronoi polygon associated to x k V k = i=1,...,n, i k {x : d(x, x k ) d(x, x i )}. Nearest-point Voronoi polygon associated to S l W l = {x : d(x, S l ) d(x, S j )}. j=1,...,m, j l

45 Nearest and farthest-point Voronoi diagrams Given the set of points {x 1,..., x n } and the set of polygons {S 1,..., S m }, Farthest-point Voronoi polygon associated to x k V k = i=1,...,n, i k {x : d(x, x k ) d(x, x i )}. Nearest-point Voronoi polygon associated to S l W l = {x : d(x, S l ) d(x, S j )}. j=1,...,m, j l

46 Nearest and farthest-point Voronoi diagrams Given the set of points {x 1,..., x n } and the set of polygons {S 1,..., S m }, Farthest-point Voronoi polygon associated to x k V k = i=1,...,n, i k {x : d(x, x k ) d(x, x i )}. Nearest-point Voronoi polygon associated to S l W l = j=1,...,m, j l {x : d(x, S l ) d(x, S j )}. The sets V = k=1,...,n V k and W = l=1,...,m W l are called the farthest-point and the nearest-point Voronoi diagrams.

47 Building the set of optimal solutions Obtain the finite dominating set of solutions (according to the necessary optimality conditions): Build the furthest-point Voronoi diagram associated to G +. Build the nearest-point Voronoi diagram associated to G. Intersection of the two diagrams. Compute the radii for any center candidate to optimal solution (see conditions 1-4). Compute the objective function in every case and obtain the optimal solution.

48 Voronoi diagrams for segments ay) for a set of 8 closed segments andlocating 2 points a Semi-Obnoxious (black). Facility Left:

49 Computational experience

50 Algorithm Our aim: Build the finite dominating set of solutions (by considering the necessary conditions for optimality). Consider all the possible combinations of active elements (for every polygon, the active element can be a vertex or an edge). Build the center x 0 as the intersection of the bisectors. Check the feasibility of the combination. Compute the radii. Compute the objective function and obtain the optimal solution.

51 Algorithm Our aim: Build the finite dominating set of solutions (by considering the necessary conditions for optimality). Consider all the possible combinations of active elements (for every polygon, the active element can be a vertex or an edge). Build the center x 0 as the intersection of the bisectors. Check the feasibility of the combination. Compute the radii. Compute the objective function and obtain the optimal solution.

52 Algorithm Our aim: Build the finite dominating set of solutions (by considering the necessary conditions for optimality). Consider all the possible combinations of active elements (for every polygon, the active element can be a vertex or an edge). Build the center x 0 as the intersection of the bisectors. Check the feasibility of the combination. Compute the radii. Compute the objective function and obtain the optimal solution.

53 Algorithm Our aim: Build the finite dominating set of solutions (by considering the necessary conditions for optimality). Consider all the possible combinations of active elements (for every polygon, the active element can be a vertex or an edge). Build the center x 0 as the intersection of the bisectors. Check the feasibility of the combination. Compute the radii. Compute the objective function and obtain the optimal solution.

54 Algorithm Our aim: Build the finite dominating set of solutions (by considering the necessary conditions for optimality). Consider all the possible combinations of active elements (for every polygon, the active element can be a vertex or an edge). Build the center x 0 as the intersection of the bisectors. Check the feasibility of the combination. Compute the radii. Compute the objective function and obtain the optimal solution.

55 Algorithm Our aim: Build the finite dominating set of solutions (by considering the necessary conditions for optimality). Consider all the possible combinations of active elements (for every polygon, the active element can be a vertex or an edge). Build the center x 0 as the intersection of the bisectors. Check the feasibility of the combination. Compute the radii. Compute the objective function and obtain the optimal solution.

56 Solution

57 Example: 20 points and 10 polygons

58 Example: 20 points and 10 polygons

59 Example: 50 points and 20 polygons

60 Example: 50 points and 20 polygons

61 Multi-instance Classification Conclusions and future work Multi-instance classification problem Circles: instances of G +1. Squares: instances of G 1. Bags: sets of instances (circles/squares) with the same colour.

62 Real database: MUSK Multi-instance Classification Conclusions and future work Bags = molecules. Instances = different conformations of low energy. Attributes = 162 measurements (distances from origin to surface) + 4 measurements (position of an oxygen atom). Labels = active or non-active. Bags G +1 Bags G 1 Total Ins G +1 Ins G 1 Total Musk Musk

63 Example (VNS algorithm) Multi-instance Classification Conclusions and future work

64 Example Multi-instance Classification Conclusions and future work

65 Example Multi-instance Classification Conclusions and future work

66 Example Multi-instance Classification Conclusions and future work

67 Example Multi-instance Classification Conclusions and future work

68 Conclusions Multi-instance Classification Conclusions and future work A model for studying the location of a semi-obnoxious facility. New techniques from Data Mining applied to Location Theory. of the problem.

69 Future work Multi-instance Classification Conclusions and future work Use of techniques from Computational Geometry to improve the algorithm. More computational experiences with real databases. Comparison between heuristics and complete enumeration. Generalization of the model to other types of balls (ellipsoids). Generalization of the model to higher dimensions. Missing values.

70 Missing values Multi-instance Classification Conclusions and future work If x ij is a missing value... x ij = x j ± kσ j, k parameter x ij = (q IQR, q IQR), inner fences where x j = mean j-th column σ j = standard deviation j-th column q 1, q 3 = quartiles IQR = interquartilic rank We are changing punctual values by intervalar values.

71 Multi-instance Classification Conclusions and future work THANKS FOR YOUR ATTENTION.