Scilab function

min_lcost_flow2 - minimum linear cost flow

### Calling Sequence

  `[c,phi,flag] = min_lcost_flow2(g)`

### Parameters

- `g` : graph list
- `c` : value of cost
- `phi` : row vector of the value of flow on the arcs
- `flag` : feasible problem flag (0 or 1)

### Description

`min_lcost_flow2` computes the minimum linear cost flow in the network `g`. It returns the total cost of the flows on the arcs `c` and the row vector of the flows on the arcs `phi`. If the problem is not feasible (impossible to find a compatible flow for instance), `flag` is equal to 0, otherwise it is equal to 1.

The bounds of the flow are given by the elements `edge_min_cap` and `edge_max_cap` 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 `edge_min_cap` or `edge_max_cap` is not given (empty row vector `[]`), it is assumed to be equal to 0 on each edge.

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

The demand on the nodes are given by the element `node_demand` 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 `node_demand` is not given (empty row vector `[]`), it is assumed to be equal to 0 on each node.

This functions uses a relaxation algorithm due to D. Bertsekas.

### Examples

    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];
    g=make_graph('foo',1,15,ta,he);
    g('node_x')=[194 191 106 194 296 305 305 418 422 432 552 550 549 416 548];
    g('node_y')=[56 221 316 318 316 143 214 321 217 126 215 80 330 437 439];
    show_graph(g);
    g1=g; ma=arc_number(g1); n=g1('node_number');
    g1('edge_min_cap')=0.*ones(1,ma);
    x_message(['Random generation of data';
               'The first(s) generated problem(s) may be unfeasible']);
    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);

### See Also

`min_lcost_cflow` `min_lcost_flow1` `min_qcost_flow`