Reviewing the integer solution
to the model,
z = 40, A1 = 1, A3 = 1, B3 = 1, B4 = 1, C1
the builder finds that the solution is impractical. He did
not spell out the requirement that a site can hold only one
building. In addition, he requires that site 2 must be used.
With 0-1 variables these constraints are easily stated:
- Site 1: A1 + B1 + C1 1
- Site 2: A2 + B2 + C2 1
- Site 3: A3 + B3 + C3 1
- Site 4: A4 + B4 + C4 1
The constraints for site 1, 3 and 4 are called mutually
exclusive constraints. If one option in a mutually exclusive
set is selected then all others must be rejected. Mathematically,
this is accomplished by restricting the sum of the 0-1 variables
in the set to be less than or equal to 1. The constraint
for site 2 is an either-or constraint. We must select
any one of the options represented in the set. These constraints
impose logical conditions on the decisions of the problem.
They are valid because the variables are forced to have only
the values 0 and 1. Notice that a requirement that at least
one of a set be selected would use a relation
instead of the .
We add the constraints to the Excel model below and once again
we solve it without the integer restrictions. Observe that
two variables are not at their bounds, where we could have
expected as many as 5, the number of constraints.
z = 39.6, A1 = 1, A2 = 1, B3 = 0.03, B4 = 1,
C3 = 0.97
The solution has design A placed on sites 1 and 2 and design
B at site 4. Site 3 has mostly design C, but partly design
B. Of course this is an impossible situation.