Project management deals with a one-time project composed of a number of activities. There are precedence relations between some of the activities that specify that one or more activities must be complete before another can begin. Activities have time durations, and the time required for an activity is known or can be estimated. Activities may use resources and generate cash flows. Before the project begins, the management wants a schedule that indicates when each activity should start and when it should be finished. While the project is ongoing, the management wants to track the progress of the activities and adjust the project schedule to accommodate unexpected occurrences.

To illustrate the model consider the following example. The same example is used in the Computations section to illustrate the Project Management add-in. This discussion uses screen shorts of Excel workbook created by that add-in. .

Structure

The first step in the analysis is to create a list of activities that comprise the project. For the example problem we identify 14 activities. We have added two more activities, one indicates the start of the project and the other end. In the list below we provide both a name and a short word description for each activity .

The story describes various ordering restrictions in terms of which activity must precede another or which activity should follow another. An obvious example is that the "Pour Slab" activity must precede the "Cure Slab" activity. These can all be described by listing predecessors for each activity as shown below for the example. Some of the activities have more than one predecessor.

We only show the immediate predecessors. Although we see that D and E precede activity H, it is also true that the predecessors of D and E, activities A and B, also precede H. We do not list A and B as predecessors of H because this relation is implied by the other entries. Although the analysis does not fail if other than immediate predecessors are listed, it is inefficient to do so. (The efficiency of the add-in is seriously impaired if very many excess relations are included.)

The table also shows successors of the activities. Whenever an activity appears in the list of predecessors, the rows in which the name appears identify the successors of the activity. For example we see C listed as predecessors of F and G. This implies that the immediate successors of C are activities F and G.

Although not shown, this table implies relations between the start and end nodes and the remainder of the activities. An activity that has no listed predecessors is an immediate successor of the start node. Similarly, an activity that has no listed successors is an immediate predecessor of the end node.

The list of activities and precedence relations describes the structure of the problem. An equivalent description of the project is the project network shown below. The activities are represented by the rectangles, called nodes. The lines connecting the nodes represent the precedence relations and are called arcs. The entries below the names are the activity times, discussed in the next paragraph, and the red colored nodes and arcs show the cortical path, to will be described later.

Time

An important characteristic of an activity is the time required from the moment the activity begins to the moment when it ends. Since the analysis takes place before the project begins, the time required for an activity must be an estimate. Of course when the activity is actually performed the time may turn out to be different than the estimate.

Most of the analysis assumes the activity time is a fixed constant. An alternative is to describe the time as a random variable with a specified probability distribution function. Traditionally, project management uses a three point estimate to describe the random variable, the minimum time, the most likely time and the maximum time. Such estimates are thought to be easier to obtain from experts or managers familiar with the tasks to be performed than estimates of the parameters of more general distributions.

More generally, any probability distribution could be be prescribed for the random variable representing time. The minimum and maximum times define the range of the distribution and the most likely time its mode. One commonly mentioned distribution is the Beta distribution.This distribution has as finite range and a mode as required by our notation, but it also has two shape parameters, alpha and beta. The shape parameters allow the model to represent a wide variety of distribution shapes. The data above does not allow the determination of the parameters for the Beta distribution, but a more general analysis could be accomplished with the help of the Random Variables add-in. This add-in provides a variety of distributions including the Beta.

Even though a model may use random variables, most of the methods to follow use a single estimate for time. That estimate is the Expected Value or Mean of the random variable. Computations involving variability use the Variance and Standard Deviation of the random variable. Traditionally, these statistical measures, or moments, are computed with the approximate formulas below.

These formulas gain in simplicity at the expense of accuracy in that they are not valid for any known general distribution and certainly not the Beta distribution. In these days of computers, there is really no need for the approximation since accurate values can easily be computed. The Random Variables add-in provides functions that compute the moments for many known distribution forms. Of course the approximation introduced by these formulas may not be important given the inherent uncertainty involved in the estimates of a, m, and b.

The figure below shows the estimated values obtained from the story and the computed moments. The column labeled Time is used for subsequent analysis. The times used are the same as the Mean values.

Resources and Cash Flows

Schedule

A principal result to obtained from our analysis is a schedule for starting the activities. This allows managers to set due dates, organize the means of production, notify suppliers on the need for resources and track progress. Generally the project has a due date and the goal is to complete all the activities before the due date. If there were no precedence relations all the activities could be performed simultaneously and the project would be complete at the time when the activity with the greatest duration is complete. If the activities could only be performed sequentially, and not simultaneously, the completion time would be the sum of the activity times.

The interesting problem comes when activities can be on-going simultaneously and precedence restricts the order of performance. This is the situation addressed in the remainder of this section. The scheduling is more complicated when there are limited resources and when the flow of cash over time is relevant. A schedule specifies a start and finish time for each activity as shown below in the last two columns. This happens to be the earliest time schedule for the project. Start and finish times are shown as numbers of periods (in this case hours) from the start of the project.

An alternative and familiar representation of the schedule is the Gantt chart shown below. Here each activity is represented by a colored bar, and the location of the bar in the chart shows the start, duration and finish of the activity.