Weber Problem Louis Luangkesorn University of Pittsburgh June 22, - - PowerPoint PPT Presentation

Weber Problem Louis Luangkesorn

Weber Problem

Louis Luangkesorn

University of Pittsburgh

June 22, 2009

Weber Problem Louis Luangkesorn

Introduction to the Weber Problem

Weber Problem: Find the "minimum" point (x∗, y∗) which minimizes the sum of weighted distances from itself to n fixed points with co-ordinate (ai.bi), where each point i has a weight wi

Weber Problem Louis Luangkesorn

Formulation of Weber Problem

min

x,y {W(x, y) = n

• t=1

widi(x, y)} di(x, y) =

• (x − ai)2 + (y − bi)2)

Weber Problem Louis Luangkesorn

Solving the Weber Problem Geometrically

• The Varignon Frame.
• This works because it is essentially the dual of the linear

programming formulation.

• The vector (ui, vi) is the negative of the force vector exerted
• n the knot by weight wi.

Weber Problem Louis Luangkesorn

Dual of the Weber problem

max

(U,V) D(U, V) = − n

• i=1

(aiui + bivi) s.t. n

i=1 ui = 0

n

i=1 vi = 0

• u2

i + v2 i ≤ wi

(1)

Weber Problem Louis Luangkesorn

Weiszfield Algorithm

• Take derivitive of the objective function and set to 0
• Start with a candidate solution
• Determine next iteration
• repeat until convergence
• Note: can have problems if a candidate location is one of the

fixed point. (Because the algorithm will only approach the fixed point.) dW(x, y) dx =

n

• i=1

wi(x − ai) di(x, y) = 0 dW(x,y)

dy=Pn

i=1 wi (y−bi ) di (x,y) =0

x(k+1), y(k+1) =  

Pn

i=1 wi ai di(x(k),y(k))

Pn

i=1 wi di (x(k),y(k))

,

Pn

i=1 wi bi di(x(k),y(k))

Pn

i=1 wi di(x(k),y(k))

  (2)

Weber Problem Louis Luangkesorn

Centroid formulation

• Minimizing sum of squared euclidean distances
• This is the centroid

min

x,y C(x, y) = n

• i=1

wid2

i (x, y)

Weber Problem Louis Luangkesorn

Rectilinear distances

Manhattan distance - only allow travel in a grid e.g. Roads

in a grid layout

min

x,y R(x, y) = n

• i=1

widR

i (x, y)

dR

i (x, y) = |x − ai| + |y − bi|

Weber Problem Louis Luangkesorn

p-norm Distance

• p-norm are generalizations of Euclidean distances

lpi =

p

• |x − ai|p + |y − bi|p

(3)

Weber Problem Louis Luangkesorn

Multiple Facilities

• Where m new facilities will be placed and one new facility y
• Weights between facility j and demand point i is wij
• Weights between facility j and demand point s is vjs

min

(x,y)j=1,...(),m n

• i=1

m

• j=1

wij

• (xj − ai)2 + (yj − bi)2+

m−1

j=1

m

s=j+1 vjs

• (xj − as)2 + (yj − bs)2

Weber Problem Louis Luangkesorn

Restricted regions

• Forbidden regions - Cannot locate in F
• minx,y F(x, y) = n

i=1 widi(x, y)

• Barriers - Path cannot cross B
• minx,y B(x, y) = n

i=1 widB i (x, y)

Weber Problem Louis Luangkesorn

Point along line

• Locate a line l which is as close as possible to the set of

points

• minlinesl L(l) = n

i=1 wiγi(l)

• γi(l) is the closest distance to point i to line l