emfl.gms : Existing Multi Facility Location Problem - Cone Format

**Description**

Euclidian multi-facility location problem using second order cone constraints. Given a set of m existing facilities, we compute the coordinates of n new facilities on a rectangular grid subject to minimizing the weighted sum of the euclidian distances between facilities. We use quadratic cone constraints to model the euclidian distances. Vanderbei, R, online at http://www.princeton.edu/~rvdb/ampl/nlmodels/facloc/emfl_socp.mod Optional inputs: --old number of existing facilities --new number of new facilities --N1 number of facilities in X direction on grid --N2 number of facilities in Y direction on grid Note that we must have new=N1*N2

**Reference**

- Vanderbei, R, Nonlinear Optimization Models (AMPL), See http://www.princeton.edu/~rvdb/ampl/nlmodels/.

**Large Model of Type :** QCP

**Category :** GAMS Model library

**Main file :** emfl.gms

```
$TITLE Existing Multi Facility Location Problem - Cone Format (EMFL, SEQ=273)
$ONTEXT
Euclidian multi-facility location problem using second order
cone constraints. Given a set of m existing facilities,
we compute the coordinates of n new facilities on a rectangular
grid subject to minimizing the weighted sum of the euclidian
distances between facilities.
We use quadratic cone constraints to model the euclidian
distances.
Vanderbei, R, online at
http://www.princeton.edu/~rvdb/ampl/nlmodels/facloc/emfl_socp.mod
Optional inputs:
--old number of existing facilities
--new number of new facilities
--N1 number of facilities in X direction on grid
--N2 number of facilities in Y direction on grid
Note that we must have new=N1*N2
$OFFTEXT
* Note that the number of new facilities must be new=N1*N2
$if not set old $set old 200
$if not set N1 $set N1 5
$if not set N2 $set N2 5
$if not set N $eval new %N1%*%N2%
Set m "old facilities" /m1*m%old%/
nX "number facilities in x direction" /nX1*nX%N1%/
nY "number facilities in y direction" /nY1*nY%N2%/
n "total number of new facilities" /n1*n%new%/
d "dimension" /"x-axis", "y-axis"/
;
Alias(nn,n);
Parameter
coords(m,d) "coordinates of existing facilities"
w(m,n) "weights associated with new-old facility pairs"
v(n,n) "weights associated with new-new facility pairs"
;
Positive Variable
x(n,d) "coordinates of new facilities"
s(m,n) "euclidian distance between new-old facilities"
t(n,n) "euclidian distance between new-new facilities"
;
Variable
diff_o(m,n,d)
diff_n(n,nn,d)
obj;
Equation
objective
diff_o_eq(m,n,d) "compute distance between new-old"
diff_n_eq(n,nn,d) "compute distance between new-new"
old_dist(m,n) "distance between new-old facilities"
new_dist(n,n) "distance between new-new facilities"
;
objective.. obj =E= sum( (m,n), w(m,n)*s(m,n)) +
sum( (n,nn), v(n,nn)*t(n,nn));
diff_o_eq(m,n,d).. diff_o(m,n,d) =E= x(n,d) - coords(m,d);
diff_n_eq(n,nn,d).. diff_n(n,nn,d) =E= x(n,d) - x(nn,d);
* Explicit cone syntax for MOSEK
*old_dist(m,n).. s(m,n) =C= sum(d, diff_o(m,n,d));
*new_dist(n,nn).. t(n,nn) =C= sum(d, diff_n(n,nn,d));
old_dist(m,n).. sqr(s(m,n)) =G= sum(d, sqr(diff_o(m,n,d)));
new_dist(n,nn).. sqr(t(n,nn)) =G= sum(d, sqr(diff_n(n,nn,d)));
Model facility /all/;
* Specify existing coordinates via uniform distribution
coords(m,d) = uniform(0,1);
* Compute weights: 0.2 for new-new facility pairs
v(n,nn)$[ord(n)<ord(nn)] = 0.2;
* Initial guess of new facility coordinates distributed evenly
* on x-y rectangle
loop((nX,nY),
loop(n$[ord(n)=( ord(nX)+card(nX)*(ord(nY)-1) )],
x.L(n,'x-axis') = (ord(nX)-0.5 )/card(nX);
x.L(n,'y-axis') = (ord(nY)-0.5 )/card(nY);
)
)
* Compute weights based on distance of coord and initial guess of
* new facility coordinates
loop((m,n),
if( abs(coords(m,'x-axis')-x.L(n,'x-axis')) <= 1/[2*card(nX)] and
abs(coords(m,'y-axis')-x.L(n,'y-axis')) <= 1/[2*card(nY)],
w(m,n) = 0.95;
elseif( abs(coords(m,'x-axis')-x.L(n,'x-axis')) <= 2/[2*card(nX)] and
abs(coords(m,'y-axis')-x.L(n,'y-axis')) <= 2/[2*card(nY)] ),
w(m,n) = 0.05;
else
w(m,n) = 0;
);
);
solve facility using qcp minimizing obj;
display x.L;
```