Again, suppose the builder looks
at the solution just obtained and finds it unacceptable. First,
he will allow design A to be specified at sites 1, 2, and/or
3 only if it is also used at site 4. Second, he wants the solution
to use all three designs. To impose these logical constraints,
we must find linear algebraic constraints that are satisfied
only if the logical conditions are true and are violated when
they are false.
The first condition is imposed by forming an *implication* constraint.
The decision to use design A at sites 1, 2 or 3 must imply
that the design A is chosen for site 4.
- Design A: A1 + A2 +A3 3*A4
or A1 + A2 +A3 - 3*A4 0.
If any one of A1, A2 or A3 is set to 1, then A4 must be 1
for feasibility. The constraint does not imply that if A4 is
selected, one of the others must also be selected, because
the inequality is satisfied if A4 is 1 and all others are 0.
We use the coefficient 3 for A4 so as not to restrict the selection
of design A for the other sites.
The second condition requires that all three designs must
be used. There are several ways to accomplish this, but here
we define three new variables.
- WA = 1 if design A is used and 0 otherwise
- WB = 1 if design B is used and 0 otherwise
- WC = 1 if design C is used and 0 otherwise
Now we add constraints that link the design decisions at the
sites to the new variables. These are *implication *constraints.
- Design A: A1 + A2 + A3 + A 4 4*WA
- Design B: B1 + B2 + B3 + B 4 4*WB
- Design C: C1 + C2 + C3 + C 4 4*WC
Now we add a constraint that requires all three designs to
be 1.
- Design Limit: WA + WB + WC = 3.
We solve the IP with the added constraints with the result
shown below. Summarizing:
z = 38, A1 = 1, A4 = 1, B3 = 1, C2 = 1.
This solution has the same objective value as the solution
obtained without the new constraints, so this is an alternative
optimum solution. It is frequently the case with IP models
that there are alternative optimum solutions. Although there
is no easy way to find them all, adding constraints that better
define the decision maker's goals may result in more acceptable
solutions. |