Project Management
- Project

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. .

The story was contributed by William Lesso, Professor Emeritus of the University of Texas.

The LongHorn Petroleum Company has just brought in a producing oil well. It’s no gusher. In fact, they need to hook up a ‘horse head’ pumping unit and run a line to a nearby set of oil storage tanks. Periodically a tanker truck will come by and take the accumulated crude oil to the refinery.

Based on past experience, we have listed the following tasks, which should be done along with estimates for this particular well of how long each task will take and the number of persons required for the task.

The first three tasks can be started anytime. At the wellhead, the ground must be cleared, forms put in and a concrete mounting slab poured. This should take about 11 hours but might be done in only 9. If there is some difficulty, it could take as long as 19 hours. Usually a crew of 4 is assigned.

The pump unit must be ordered. This should take one person only a few minutes, but if he has difficulty finding a supplier with one might take an hour.

A path must be cut through the brush and cactus from the well to the storage tanks. This should take 30 hours. It may only require 20 hours if the brush isn’t too thick, but it could take as long as 40. A crew of six will be assigned to this task.

Once the concrete slab is poured, it must cure for 24 hours, no more, no less.

After the order is placed, the pumping unit will be delivered to the well site by the supply company in 9 hours. The time will be no more than that, but the unit may arrive in only 3 hours.

After the path is cleared, a crew of six will dig a trench, put in a flow line and cover it. This usually takes at least 10 hours. It could be as long as 14, but the task should be done in 11.

At the same time, three persons will run an electric line along the path. They estimate a likely time of 10 hours, but it may be as long as 12 and as short as 8.

Once the pumping unit has been delivered and the cement slab has cured, it will take a crew of three persons 5 hours to mount the unit but it might be finished in only 4. If things don’t work out well, it could take as long as 12 hours.

With the pump unit in place, two persons will hook up an electric motor. This will take a minimum of 3 hours and as much as 5, but mostly likely 4.

At the same time, three persons will run sucker rods down the well and to the pump. Thus usually takes 8 hours but might only take half as long. If there are problems, it could take 18 hours.

When the electric line has been run and the electric motor is in place, it will be tied in to the electric supply and tested. One person can accomplish this job in 1 hour.

When the flow line has been completed and the sucker rods installed, the well must be tied in and all the valves ‘nippled up.’ It takes two persons an hour to do this. The time can be no less than an hour, and it could be 4 hours if there are problems.

After the electric motor is ready and the well has been tied into the flow line, the final test of the pump unit is made. This usually takes about 4 hours but could be done within 1 hour. However, it has been known to take as much as 10 hours if many adjustments need to be made. A crew of two does this job.

At the same time, one person goes around and makes sure all the values are open. This could take only a few minutes. More likely it could take an hour, but never more that 2.

With these tasks completed, the unit is ready to pump oil.





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.

  This form of the network is called the actvity-on-node network. Another form is called the activity-on-arc network. We choose the former because it is easier to construct and is used by the Project Management add-in. The action of constructing this network is often call networking, and it is an important part of project management.



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

  The table shows one column labeled resources, in this case named Crew. Activities may be associated with several different resources and also cash flows as described on following pages. The principal assumption regarding resources is that they only be required during the time the activity is being performed and that the amount required is constant during that period. Thus, activity A requires a crew of 4 persons. That number is to be available during the entire 12 hours during the pouring operation.




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.

  The methods discussed in the remainder of this section address this scheduling problem.



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Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved