$ontext

  Alphametics, stable formulation

              AN
  + ACCELERATING
   + INFERENTIAL
   + ENGINEERING
          + TALE
         + ELITE
         + GRANT
           + FEE
            + ET
        + CETERA =
   -------------
      ARTIFICIAL
  + INTELLIGENCE

   Erwin Kalvelagen, 2009

$offtext

set
   i 'letters' /A,N,C,E,L,R,T,I,G,F/
   w 'words'   /an,accelerating,inferential,engineering,tale,elite,grant,fee,et,cetera,
                artificial,intelligence/
   w1(w)       /an,accelerating,inferential,engineering,tale,elite,grant,fee,et,cetera/
   w2(w)       /artificial,intelligence/
;

alias (i,j);
abort$(card(i) <> 10) "set i should have 10 elements";

set
  k 'slices (slice1 if rightmost slice)' /slice1*slice12/
  lead(i) 'leading letters' /A,I,E,T,G,F,C/
;

set letters(w,i,k) /
an.(a.slice2, n.slice1)
accelerating.(a.slice12, c.slice11, c.slice10, e.slice9, l.slice8, e.slice7,
   r.slice6, a.slice5, t.slice4, i.slice3, n.slice2, g.slice1)
inferential.(i.slice11, n.slice10, f.slice9, e.slice8, r.slice7, e.slice6,
   n.slice5, t.slice4, i.slice3, a.slice2, l.slice1)
engineering.(e.slice11, n.slice10, g.slice9, i.slice8, n.slice7, e.slice6,
   e.slice5, r.slice4, i.slice3, n.slice2, g.slice1)
tale.(t.slice4, a.slice3, l.slice2, e.slice1)
elite.(e.slice5, l.slice4, i.slice3, t.slice2, e.slice1)
grant.(g.slice5, r.slice4, a.slice3, n.slice2, t.slice1)
fee.(f.slice3, e.slice2, e.slice1)
et.(e.slice2, t.slice1)
cetera.(c.slice6, e.slice5, t.slice4, e.slice3, r.slice2, a.slice1)
artificial.(a.slice10, r.slice9, t.slice8, i.slice7,
   f.slice6, i.slice5, c.slice4, i.slice3, a.slice2, l.slice1)
intelligence.(i.slice12, n.slice11, t.slice10, e.slice9, l.slice8, l.slice7,
   i.slice6, g.slice5, e.slice4, n.slice3, c.slice2, e.slice1)

/;

parameter lhs(k,i), rhs(k,i);
lhs(k,i) = sum(letters(w1,i,k),1);
rhs(k,i) = sum(letters(w2,i,k),1);

parameter v(i); v(i) = ord(i)-1;

variables
  y(i)     'decision variables'
  x(i,j)   'auxiliary variables for uniqueness'
  carry(k)
  z        'dummy objective variable'
;
binary variable x;
integer variable carry;

equation
  addition(k)  'the actual problem'
  ydef(i)      'calculate y'
  xrow(i)      'row sums'
  xcol(j)      'column sums'
  dummy_objective

;


addition(k)..
   carry(k-1) + sum(i, lhs(k,i)*y(i)) =e=
       sum(i, rhs(k,i)*y(i)) + 10*carry(k);

carry.fx('slice12')=0;

ydef(i).. y(i) =e= sum(j, x(i,j)*v(j));
xrow(i).. sum(j, x(i,j)) =e= 1;
xcol(j).. sum(i, x(i,j)) =e= 1;

*
* bounds on leading digits: they can not be 0
*
y.lo(lead) = 1;

dummy_objective.. z =e= sum(i, y(i));


model m /all/;
solve m using mip minimizing z;
option y:0;
display y.l;