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 <a href="http://www.princeton.edu/~rvdb/ampl/nlmodels/facloc/emfl_socp.mod">http://www.princeton.edu/~rvdb/ampl/nlmodels/facloc/emfl_socp.mod</a> 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;