.TH min_lcost_flow2 1 "September 1995" "Scilab Group" "Scilab function"
.so man1/sci.an
.SH NAME
min_lcost_flow2 - minimum linear cost flow
.SH CALLING SEQUENCE
.nf
[c,phi,flag]=min_lcost_flow2(g)
.fi
.SH PARAMETERS
.TP 2
g
: graph-list
.TP 2
c 
: value of cost
.TP 4
phi
: row vector of the value of flow on the arcs
flag
: feasible problem flag (0 or 1)
.SH DESCRIPTION
\fVmin_lcost_flow2\fR computes the minimum linear cost flow in the network 
\fVg\fR. It returns the total cost of the flows on the arcs \fVc\fR and
the row vector of the flows on the arcs \fVphi\fR. If the problem is not 
feasible (impossible to find a compatible flow for instance), \fVflag\fR is 
equal to 0, otherwise it is equal to 1.

The bounds of the flow are given by the elements \fVedge_min_cap\fR and
\fVedge_max_cap\fR of the graph-list. 
The value of the minimum capacity must be equal to zero. The values of the 
maximum capacity must be non negative and must be integer numbers.
If the value of \fVedge_min_cap\fR or \fVedge_max_cap\fR is not given (empty
row vector \fV[]\fR), it is assumed to be equal to 0 on each edge.

The costs on the edges are given by the element \fVedge_cost\fR of the 
graph-list.
The costs must be non negative and must be integer numbers.
If the value of \fVedge_cost\fR is not given (empty row vector \fV[]\fR), 
it is assumed to be equal to 0 on each edge.

The demand on the nodes are given by the element \fVnode_demand\fR of the 
graph-list. 
The demands must be integer numbers. Note that the sum of the demands must
be equal to zero for the problem to be feasible.
If the value of \fVnode_demand\fR is not given (empty row vector \fV[]\fR), 
it is assumed to be equal to 0 on each node.

This functions uses a relaxation algorithm due to D. Bertsekas.
.SH EXAMPLE
.nf
ta=[1 1 2 2 2 3 4 4 5 6 6 6 7 7 7 8 9 10 12 12 13 13 13 14 15 14 9 11 10 1 8];
he=[2 6 3 4 5 1 3 5 1 7 10 11 5 8 9 5 8 11 10 11 9 11 15 13 14 4 6 9 1 12 14];
gt=make_graph('foo',1,15,ta,he);
gt('node_x')=[194 191 106 194 296 305 305 418 422 432 552 550 549 416 548];
gt('node_y')=[56 221 316 318 316 143 214 321 217 126 215 80 330 437 439];
show_graph(gt,'rep');
g1=gt; ma=g1('arc_number'); n=g1('node_number');
g1('edge_min_cap')=0.*ones(1,ma);
while %T then
 rand('uniform');
 g1('edge_max_cap')=round(20*rand(1,ma))+20*ones(1,ma);
 g1('edge_cost')=round(10*rand(1,ma)+ones(1,ma));
 rand('normal');
 dd=20.*rand(1,n)-10*ones(1,n);
 dd=round(dd-sum(dd)/n*ones(1,n));
 dd(n)=dd(n)-sum(dd);
 g1('node_demand')=dd;
 [c,phi,flag]=min_lcost_flow2(g1);
 if flag==1 then break; end;
end;
x_message(['The cost is: '+string(c);
           'Showing the flow on the arcs and the demand on the nodes']);
ii=find(phi<>0); edgecolor=phi; edgecolor(ii)=11*ones(ii);
g1('edge_color')=edgecolor;
edgefontsize=8*ones(1,ma); edgefontsize(ii)=18*ones(ii);
g1('edge_font_size')=edgefontsize;
g1('edge_label')=string(phi);
g1('node_label')=string(g1('node_demand'));
show_graph(g1,'rep');
.fi
.SH SEE ALSO
min_lcost_cflow, min_lcost_flow1, min_qcost_flow