Lecture No 20

 

Held on: Monday, October 9, 2000

 

Notes Prepared by: Samarth  Chandra & Rajesh Das (97262)

 

 

Facility Location Problem

Issue is where to locate a new facility.

(x_{i ,} y_i )   ,i = 1,2,…..m.

w_i = cost per unit distance to serve facility i.

Let say we know this.

Let the location of the facility is X (x, y).

Cost of facility is S( i=1 to m)  w_i((x- x_i)²+(y- y_i)²) ^½

On differentiating:

             ¶C(x,y)/¶x = 0

             ¶C(x,y)/¶y = 0

on solving these eqns we get

   S( i=1 to m)  w_i(x- x_i) /((x- x_i)²+(y- y_i)²) ^½  = 0

   S( i=1 to m)  w_i(y- y_i) /((x- x_i)²+(y- y_i)²) ^½  = 0

If each w_i = constant K,

X= 1/m S( i=1 to m) x_i

Y= 1/m S( i=1 to m) y_i

C(x,y) =   S( i=1 to m)  w_i(lx- x_il+ly- y_il)

min C(x,y)

find x that minimize

S( i=1 to m)  w_i(lx- x_il) ,

S( i=1 to m)  w_i(ly- y_il)

Let us take an example.

 

min C(x)=åw(i)|X-X(i)|

Rewrite x1,x2…..xm in terms of unique order statistics as:

x(1),x(2)….x(m) where x(i)<x(j) wherever i<j.   m’<m

w(1),w(2)…..w(m)

C(x)=

C’(x)=-

For x(t)<x<x(t+1)

Suppose

…………….(2)

at some t=t*.

If (2) is an equality then

     X*  e  [X(t*),X(t*+1)]