$title Peacefully Coexisting Armies of Queens (COEX,SEQ=219)

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Two armies of queens (black and white) peacefully coexist on a
chessboard when they are placed on the board in such a way that
no two queens from opposing armies can attack each other. The
problem is to find the maximum two equal-sized armies.

Bosch, R, Mind Sharpener. OPTIMA MPS Newsletter (2000).

Keywords: mixed integer linear programming, mathematical games, combinatorial optimization,
          peaceably coexisting armies of queens
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Set i 'size of chess board' / 1*8 /;

Alias (i,j,ii,jj);

Set M(i,j,ii,jj) 'shared positions on the board';

M(i,j,ii,jj) = (ord(i) = ord(ii)) or
               (ord(j) = ord(jj)) or
               (abs(ord(i) - ord(ii)) = abs(ord(j) - ord(jj)));

Binary Variable
   b(i,j) 'square occupied by a black queen'
   w(i,j) 'square occupied by a white queen';

Variable tot 'total queens in each army';

Equation
   eq1(i,j,ii,jj) 'keeps armies at peace'
   eq2            'add up all the black queens'
   eq3            'add up all the white queens';

eq1(m(i,j,ii,jj)).. b(i,j) + w(ii,jj) =l= 1;

eq2.. tot =e= sum((i,j), b(i,j));

eq3.. tot =e= sum((i,j), w(i,j));

Model armies / all /;

option limCol = 0, limRow = 0;

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* solution reported in OPTIMA
b.fx('6','2') = 1;
b.fx('7','2') = 1;
b.fx('8','2') = 1;
b.fx('7','1') = 1;
b.fx('8','1') = 1;
b.fx('7','6') = 1;
b.fx('8','6') = 1;
b.fx('7','7') = 1;
b.fx('8','7') = 1;
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solve armies maximizing tot using mip;