In this example, we will introduce the use of a sparse derived set with an explicit listing. As you recall, when we use this technique to define a sparse set, we must explicitly list all members belonging to the set. This will usually be some small subset of the dense set resulting from the full Cartesian product of the parent sets.

For our example, we will set up a PERT (Project Evaluation and Review Technique) model to determine the critical path of tasks in a project involving the roll out of a new product. PERT is a simple, but powerful, technique developed in the 1950s to assist managers in tracking the progress of large projects. PERT is particularly useful in identifying the critical activities within a project, which, if delayed, will delay the project as a whole. These time critical activities are referred to as the critical path of a project. Having such insight into the dynamics of a project goes a long way in guaranteeing it won't get sidetracked and become delayed. In fact, PERT proved so successful, the Polaris project that it was first used on was completed 18 months ahead of schedule. PERT continues to be used successfully on a wide range of projects. For more information on PERT, and a related technique called CPM (Critical Path Method), please refer to Schrage (2005) or Winston (1995).

The formulation for this model is included in the SAMPLES subdirectory off the main LINGO directory under the name PERT.