MATH 441 Section 201
Sample inputs for LINGO for Fano Plane example.
Online Course Material

LINGO is a higher level language from which you can use Linear programming. The LP model is typically formulated in more general terms with access to data files containing for example, the requirements or availablities of raw materials and sets of variables etc. Examples found in the HELP function may be useful. We can define any projective plane of order n as a n^2+n+1 by \$n^2+n+1 (0,1)-matrix with row and columns sums n+1 and such that there is no 2x2 submatrix
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Thus the Fano Plane as a 7x7 (0,1)-matrix such tha teach row has row sum 3 and each column has column sum 3 and there is no 2 by 2 submatrix of 1's. We can define this search as a integer linear program.
Model:
!x(i,j)=1 if element j is in block i;
! we first set up sets block={1,2,3,4,5,6,7} and element={1,2,3,4,5,6,7};
!then we form a derived set entry of the Cartesian product of block and
element; !with x being a characteristic of the derived set entry so that;
!we have defined the array x(i,j);

Sets:
block /1..7/;
element /1..7/;
entry (block,element): x;
endsets
Max=x(1,1);
@for(block(i): @sum(element(j): x(i,j))=3);
@for(element(i): @sum(block(j): x(i,j))=3);
@for(block(i): @for(block(j) | i#NE#j: @for(element(k):
@for(element(l) | k#NE#l: x(i,k)+x(i,l)+x(j,k)+x(j,l) < 3))));
@for(entry: @GIN(x));

end