## Mathematical Programming

Mathematical programming 116, 18, 41, 551 can be used to obtain optimal solutions to certain types of multiproject scheduling problems. These procedures determine when an activity should be scheduled, given resource constraints. In the following discussion, it is important to remember that each of the techniques can be appliĀ®? to the activities in a single project, or to the projects in a partially or wholly interde; pendent set of projects. Most models are based on integer programming that formulates the problem using 0-1 variables to indicate (depending on task early start times, due dates, sequencing relationships, etc,) whether or not an activity is schefl' uled in specific periods. The three most common objectives are these:

1. Minimum total throughput time (time in the shop) for all projects

2. Minimum total completion time for all projects

3. Minimum total lateness or lateness penalty for all projects.

Constraint equations ensure that every schedule meets any or all of the following constraints, given that the set of constraints allow a feasible solution.

1. Limited resources

2. Precedence relationships among activities

3. Activity-splitting possibilities

4. Project and activity due dates

5. Substitution of resources to assign to specified activities

6. Concurrent and nonconcurrent activity performance requirements

In spite of its ability to generate optimal solutions, mathematical programming has some serious drawbacks when used for resource allocation and multiproject scheduling. As noted earlier, except for the case of small problems, this approach has proved to be extremely difficult and computationally expensive.