How to specify an IF-THEN-ELSE constraint with an Integer Linear Programming (ILP) solver

Problem :

    For example:  if (P = 1) then A < B else A > B.    (Note that P is a binary variable, that is, P = {0, 1}.)

Answer :

if (P = 1) then A < B else A > B.
==> 
          if (P = 1) then A < B else B < A.

==> 
          if (P = 1) then (A - B < 0) else (B - A < 0).

==>
          A - B < M1*(1-P)    .......... (1)
          B - A < M2*P          .......... (2)

          Note that P is a binary variable, that is, P = {0, 1}. In addition, M1 and M2 are constants whose values should be large enough.
          If P=1, then A < B and constraint (2) is redundant.
          If P=0, then B < A and constraint (1) is redundant.

==>
          A - B <= M1*(1-P) - 1        .......... (3)
          B - A <= M2*P - 1              .......... (4)

          Note that "<=" denotes "less than or equal". 

last update: May 29, 2009