Models

Site Selection

 The Problem Solving the Model Logical Constraints More Logical Constraints
 Site Selection Logical Constraints
 Reviewing the integer solution to the model, z = 40, A1 = 1, A3 = 1, B3 = 1, B4 = 1, C1 = 1, 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. Re-imposing the integrality requirement, we obtain quite a different answer. z = 38, A1 = 1, A2 = 1, B3 = 1, C4 = 1. An alternative optimum uses only two designs and three sites. z = 38, A2 = 1, C1 = 1, C3 = 1.

Operations Research Models and Methods
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by Paul A. Jensen