Activity Earliest and Latest Times and Slack

From the table, we can see that the critical path encompasses activities 2-5-8-11, since these activities have no available slack. We can also see that the expected project completion time (t_p) is the same as the earliest or latest finish for activity 11, or t_p = 25 weeks. To determine the project variance, we sum the variances for the activities on the critical path. Using the variances shown in the table for the critical path activities, we can compute the total project variance as follows:

ALONG THE SUPPLY CHAIN An Interstate Highway Construction Project in Virginia

One of the most frequent applications of project management is for construction projects. Many companies and government agencies contractually require a formal project management process plan as part of the bid process. One such project is for the construction of four new high occupancy toll (HOT) lanes totaling 96 miles on 1-495, one of the nation's busiest traffic corridors near Washington, DC, at a cost of $1.4 billion. HOT lanes are tolled lanes that operate alongside existing highways to provide drivers with a faster and more reliable travel option, for a fee. This Virginia Department of Transportation (VDOT) five-year project scheduled to be completed in 2013 also includes replacing more than 50 aging bridges and overpasses, upgrading 10 interchanges, improving bike and pedestrian access, and improving sound protection for local neighborhoods. The project employs as many as 500 skilled workers on-site that work under extreme safety conditions. As many as a quarter million vehicles pass through this corridor daily so all work is done under heavy traffic conditions. Much of it is done at night, during off-hours, and on weekends, which adds costs for premium-time and late-shift pay and makes scheduling complex and difficult. Another dimension which has added complexity to the project is that it requires a “public-private” partnership between VDOT and private contractors. While VDOT, as a government agency, has a responsibility to review and evaluate every contract package carefully (and sometimes slowly), the private contractors want to accelerate the contract biding and award process in order to maintain schedules. When designs have not been approved and contracts have not been bid, the contractors may not be able to keep workers busy and work will slow down and lag. Thus, the project management process requires a high degree of communication and coordination between team members, and focused administration by the project leaders.

Source: Sarah Gale, “A Closer Look: Virginia Department of Transportation,” PM Network, vol. 23 (4: April 2009), pp. 48-51; and, the Virginia Department of Transportation Web site at http://virginiadot.org.

CPM/PERT ANALYSIS WITH OM TOOLS

The “Project Management” module in OM Tools has the capability to develop both single-time estimate and three-time estimate networks. Exhibit 9.1 shows the OM Tools spreadsheet for the “Order Processing System” project in Example 9.1.

Exhibit 9.1

• OM Tools File

PROBABILISTIC NETWORK ANALYSIS

The CPM/PERT method assumes that the activity times are statistically independent, which allows us to sum the individual expected activity times and variances to get an expected project time and variance. It is further assumed that the network mean and variance are normally distributed. This assumption is based on the central limit theorem of probability, which for CPM/PERT analysis and our purposes states that if the number of activities is large enough and the activities are statistically independent, then the sum of the means of the activities along the critical path will approach the mean of a normal distribution. For the small examples in this chapter, it is questionable whether there are sufficient activities to guarantee that the mean project completion time and variance are normally distributed. Although it has become conventional in CPM/PERT analysis to employ probability analysis using the normal distribution regardless of the network size, the prudent user should bear this limitation in mind.

Probabilistic analysis of a CPM/PERT network is the determination of the probability that the project will be completed within a certain time period given the mean and variance of a normally distributed project completion time. This is illustrated in Figure 9.14. The value Z is computed using the following formula:

Figure 9.14 Normal Distribution of Project Time

where

This value of Z is then used to find the corresponding probability in Table A.1 (Appendix A).

Example 9.2 Probabilistic Analysis of the Project Network

The Southern Textile Company in Example 9.1 has told its customers that the new order processing system will be operational in 30 weeks. What is the probability that the system will be ready by that time?

The probability that the project will be completed within 30 weeks is shown as the shaded area in the accompanying figure. To compute the Z value for a time of 30 weeks, we must first compute the standard deviation (σ) from the variance (σ²).

Next we substitute this value for the standard deviation along with the value for the mean, 25 weeks, and our proposed project completion time, 30 weeks, into the following formula:

A Z value of 1.91 corresponds to a probability of 0.4719 in Table A.1 in Appendix A. This means that there is a 0.9719 probability of completing the project in 30 weeks or less (adding the probability of the area to the left of μ = 25, or 0.5000 to 0.4719).

Example 9.3 Probabilistic Analysis of the Project Network

A customer of the Southern Textile Company has become frustrated with delayed orders and told the company that if the new ordering system is not working within 22 weeks, it will not do any more business with the textile company. What is the probability the order processing system will be operational within 22 weeks?

The probability that the project will be completed within 22 weeks is shown as the shaded area in the accompanying figure.

The probability of the project's being completed within 22 weeks is computed as follows:

A Z value of −1.14 corresponds to a probability of 0.3729 in the normal table in Appendix A. Thus, there is only a 0.1271 (i.e., 0.5000 − 0.3729) probability that the system will be operational in 22 weeks.

MICROSOFT PROJECT

• Microsoft Project File

When you open Microsoft Project, a screen comes up for a new project. Click on the “Tasks” button and the screen like the one shown in Exhibit 9.2 will appear. Notice the set of steps on the left side of the screen starting with “define the project.” If you click on “this step,” it enables you to set a start date and save it. We set the start date for our house building project as June 10, 2010.

Exhibit 9.2

The second step in the “Tasks” menu. “Define general working times,” allows the user to specify general work rules and a work calendar including such things as working hours per day, holidays, and weekend days off. In the third step in the “Tasks” menu, we can “List the tasks in the project.” The tasks for our house building project are shown in Exhibit 9.3. Notice that we also indicated the duration of each task in the “Duration” column. For example, to enter the duration of the first activity, you would type in “3 months” in the duration column. Notice that the first task is shown to start on June 10.

The next thing we will do is “Schedule tasks” by specifying the predecessor and successor activities in our network. This is done by using the buttons under the “Link dependent tasks” window in Exhibit 9.3. For example, to show that activity 1 precedes activity 2, we put the cursor on activity 1 and then hold down the “Ctrl” key while clicking on activity 2. This makes a “finish to start” link between two activities, which means that activity 2 cannot start until activity 1 is finished. This creates the precedence relationship between these two activities, which is shown under the “Predecessor” column in Exhibit 9.3.

Exhibit 9.3

Exhibit 9.4

To see the project network, click on the “View” button again on the toolbar and then click on “Network Diagram.” Exhibit 9.4 shows the project network with the critical path highlighted in red. Exhibit 9.5 shows the project network “nodes” using the “Zoom” option from the “View” menu to increase the network size. Notice that each node includes the start and finish dates and the activity number.

Exhibit 9.5

Exhibit 9.6

Exhibit 9.7

PERT ANALYSIS WITH MICROSOFT PROJECT

A PERT network with three time estimates can also be developed using Microsoft Project. We will demonstrate this capability using our “Order Processing System” project from Example 9.1. After all of the project tasks are listed, then the three activity time estimates are entered by clicking on the “PERT Entry Sheet” button on the toolbar. If this button is not on your toolbar, you must add it in by clicking on “View” and then from the Toolbars options select “PERT Analysis.” Exhibit 9.8 shows the PERT time estimate entries for the activities in our order processing system project example. Exhibit 9.9 shows the estimated activity durations based on the three time estimates for each activity, the precedence relationships, and the project Gantt chart. Exhibit 9.10 shows the project network.

Exhibit 9.8

Exhibit 9.9

Exhibit 9.10

ALONG THE SUPPLY CHAIN The Corps of Engineers Hurricane Katrina New Orleans Restoration Project

Although category 3 Hurricane Katrina veered east of New Orleans and the city avoided a direct hit on August 29. 2005, the resulting rising floodwaters from the Mississippi River crushed the city's levees and floodwalls, causing the flood protection system to fail at more than 50 places and sub-merging 80% of the city. Almost 2,000 people lost their lives and damage estimates totaled well over $100 billion, making it arguably the largest natural disaster in the history of The United States Less than a month later Hurricane Rita, the third-largest storm in U.S. history, hit the same Gulf coast area.

It was the responsibility of the United States Army Corps of Engineers to help the city recover and restore the flood protection system in a very short period of time. The project task list included pumping or draining 250 billion gallons of water from New Orleans; removing about 28 million cubic yards of debris from the city and Gulf Coast; and repairing and restoring 220 miles of levees and floodwalls to at least pre-Katrina levels. The project team was also charged with rebuilding floodgates, upgrading the hurricane protection system in New Orleans, and restoring navigation along the Mississippi River. Draining floodwater was a priority since it was essential for public health; the original estimate was that it would take six months but it was accomplished in 45 days. At the same time Army helicopter crews working around the clock dropped an average of 600 sandbags weighing 7,000 pounds every day for 10 days and closed the system's breaches only two weeks after the hurricane hit. Meanwhile debris was removed, water distributed (including 170 million pounds of ice), power restored, over 81,000 temporary roofs were installed, 80.000 trees were removed, and, 7,100 structures were demolished. The levees were rebuilt with erosion-resistant clay and a more stable floodwall formation by mid-January 2006, well before the target date of June 1, the start of the next hurricane season. The project team accomplished what might normally have taken a decade in eight months. The project cost $3.7 billion in Louisiana alone, and more than 10,000 workers were involved with the project including 8,000 Corps of Engineers employees and team members from as far away as the Netherlands, Germany, Korea, and Japan. Despite public criticism of the Federal Emergency Management Agency (FEMA). the city's preparedness, and the government's handling of various recovery and citizen relocation efforts, the Corps of Engineer's restoration project immediately following the hurricane was a success by any measure.

PROJECT CRASHING AND TIME-COST TRADEOFF

The project manager is frequently confronted with having to reduce the scheduled completion time of a project to meet a deadline. In other words, the manager must finish the project sooner than indicated by the CPM/PERT network analysis. Project duration can often be reduced by assigning more labor to project activities, in the form of overtime, and by assigning more resources (material, equipment, and so on). However, additional labor and resources increase the project cost. Thus, the decision to reduce the project duration must be based on an analysis of the tradeoff between time and cost. Project crashing is a method for shortening the project duration by reducing the time of one (or more) of the critical project activities to less than its normal activity time. This reduction in the normal activity time is referred to as crashing. Crashing is achieved by devoting more resources, usually measured in terms' of dollars, to the activities to be crashed.

Crashing: reducing project time by expending additional resources.

Figure 9.15 The Project Network for Building a House

PROJECT CRASHING

To demonstrate how project crashing works, we will employ the CPM/PERT network for constructing a house in Figure 9.8. This network is repeated in Figure 9.15, except that the activity times previously shown as months have been converted to weeks. Although this sample network encompasses only single-activity time estimates, the project crashing procedure can be applied in the same manner to PERT networks with probabilistic activity time estimates.

We will assume that the times (in weeks) shown on the network activities are the normal activity times. For example, 12 weeks are normally required to complete activity 1. Furthermore, we will assume that the cost required to complete this activity in the time indicated is $3000. This cost is referred to as the normal activity cost. Next, we will assume that the building contractor has estimated that activity 1 can be completed in seven weeks, but it will cost S5000 instead of $3000 to complete the activity. This new estimated activity time is known as the crash time, and the cost to achieve the crash time is referred to as the crash cost.

Activity 1 can be crashed a total of five weeks (normal time − crash time = 12-7 = 5 weeks) at a total crash cost of $2000 (crash cost − normal cost = $5000 − 3000 = $2000). Dividing the total crash cost by the total allowable crash time yields the crash cost per week:

Total crash cost\Total crash time = $2000/5 = $400 per week

If we assume that the relationship between crash cost and crash time is linear, then activity 1 can be crashed by any amount of time (not exceeding the maximum allowable crash time) at a rate of $400 per week. For example, if the contractor decided to crash activity 1-2 by only two weeks (reducing activity time to 10 weeks), the crash cost would be $800 ($400 per week × 2 weeks). The linear relationships between crash cost and crash time and between normal cost and normal time are illustrated in Figure 9.16.

Crash time: an amount of time an activity is reduced.

Crash cost: is the cost of reducing activity time.

Figure 9.16 The Relationship Between Normal Time and Cost, and Crash Time and Cost

The goal of crashing is to reduce project duration at minimum cost.

The objective of project crashing is to reduce project duration while minimizing the cost of crashing. Since the project completion time can be shortened only by crashing activities on the critical path, it may turn out that not all activities have to be crashed. However, as activities are crashed, the critical path may change, requiring crashing of previously noncritical activities to reduce the project completion time even further.

Example 9.4 Project Crashing

Recall that the critical path for the house building network in Figure 9.15 encompassed activities 1-2-7 and the project duration was nine months, or 36 weeks. Suppose the home builder needed the house in 30 weeks and wanted to know how much extra cost would be incurred to complete the house by this time.

The normal times and costs, the crash times and costs, the total allowable crash times, and the crash cost per week for each activity in the network in Figure 9.15 are summarized in the following table:

We start by looking at the critical path and seeing which activity has the minimum crash cost per week. Observing the preceding table and the figure below, we see activity 1 has the minimum crash cost of $400. Activity 1 will be reduced as much as possible. The table shows that the maximum allowable reduction for activity 1 is five weeks, but we can reduce activity 1 only to the point at which another path becomes critical. When two paths simultaneously become critical, activities on both must be reduced by the same amount. If we reduce the activity I time beyond the point at which another path becomes critical, we may be incurring an unnecessary cost. This last stipulation means that we must keep up with all the network paths as we reduce individual activities, a condition that makes manual crashing very cumbersome. For that reason the computer is generally required for project crashing: however, we will solve this example manually in order to demonstrate the logic of project crashing.

It turns out that activity 1 can be crashed by the total amount of five weeks without an- other path becoming critical, since activity 1 is included in all four paths in the network.

Crashing this activity results in a revised project duration of 31 weeks at a crashing cost of S2000. The revised network is shown in the following figure.

Since we have not reached our crashing goal of 30 weeks, we must continue, and the process is repeated. The critical path in the preceding figure remains the same, and the minimum activity crash cost on the critical path is $500 for activity 2. Activity 2 can be crashed a total of three weeks, but since the contractor desires to crash the network only to 30 weeks, we need to crash activity 2 by only one week. Crashing activity 2 by one week does not result in any other path becoming critical, so we can safely make this reduction. Crashing activity 2 to seven weeks (i.e., a one-week reduction) costs $500 and reduces the project duration to 30 weeks.

The total cost of crashing the project to 30 weeks is $2500. The contractor could inform the customer that an additional cost of only $2500 would be incurred to finish the house in 30 weeks.

Suppose we wanted to continue to crash this network, reducing the project duration down to the minimum time possible—that is, crashing the network the maximum amount possible. We can determine how much the network can be crashed by crashing each activity the maximum amount possible and then determining the critical path of this completely crashed network. For example, activity 1 is seven weeks, activity 2 is five weeks, 3 is three weeks, and so on. The critical path of this totally crashed network is 1-2-4—6-7, with a project duration of 24 weeks. This is the least amount of time in which the project can be completed. If we crashed all the activities by their maximum amount, the total crashing cost would be $35,700, computed by subtracting the total normal cost of $75,000 from the total crash cost of $110,700 in the preceding table. However, if we followed the crashing procedure outlined in this example, the network could be crashed to 24 weeks at a cost of $31,500, a savings of $4000.

THE GENERAL RELATIONSHIP OF TIME AND COST

In our discussion of project crashing, we demonstrated how the project critical path time could be reduced by increasing expenditures for labor and other direct resources. The objective of crashing was to reduce the scheduled completion time to reap the results of the project sooner. However, there may be other reasons for reducing project time. As projects continue over time, they consume indirect costs, including the cost of facilities, equipment, and machinery, interest on investment, utilities, labor, personnel costs, and the loss of skills and labor from members of the project team who are not working at their regular jobs. There also may be direct financial penalties for not completing a project on time. For example, many construction contracts and government contracts have penalty clauses for exceeding the project completion date.

Figure 9.17 The Time-Cost Tradeoff

In general, project crashing costs and indirect costs have an inverse relationship; crashing costs are highest when the project is shortened, whereas indirect costs increase as the project duration increases. This time—cost relationship is illustrated in Figure 9.17. The best, or optimal, project time is at the minimum point on the total cost curve.

SUMMARY

CPM/PERT has also been widely applied in the private sector. Two of the areas of application of CPM/PERT in the private sector have been research and development (R&D) and construction. CPM/PERT has been applied to R&D projects, such as developing new drugs, planning and introducing new products, and developing new and more powerful computer systems. CPM/PERT analysis has been particularly applicable to construction projects. Almost every type of construction project—from building a house to constructing a major sports stadium, to building a ship, to constructing the Alaska oil pipeline—has been the subject of network analysis.

One reason for this popularity is that a network analysis provides a visual display of the project that is easy for managers and staff to understand and interpret. It is a powerful tool for identifying and organizing the activities in a project and controlling the project schedule. However, beyond that it provides an effective focal point for organizing the efforts of management and the project team.

Currently, hundreds of project management software packages are commercially available for personal computers, ranging in cost from several hundred dollars to thousands of dollars.

CPM/PERT also has certain limitations. The project manager tends to rely so heavily on the project network that errors in the precedence relationship or missing activities can be overlooked, until a point in time where these omissions become a problem. Attention to critical path activities can become excessive to the extent that other project activities may be neglected, or they may be delayed to the point that other paths become critical. Obtaining accurate single-time estimates and even three probabilistic time estimates is difficult and subject to a great deal of uncertainty. Since persons directly associated with the project activity within the organization are typically the primary source for time estimates, they may be overly pessimistic if they have a vested interest in the scheduling process or overly optimistic if they do not. Personal interests aside, it is frequently difficult to define, within the context of an activity, what an optimistic or pessimistic time means. Nevertheless, such reservations have not diminished the popularity of CPM/PERT because most people feel its usefulness far outweighs any speculative or theoretical drawbacks.

SUMMARY OF KEY FORMULAS

ES = max (EF of immediately preceding activities)

EF = ES + t

Latest Start and Finish Times

LS = LF - t

LF = min (LS of immediately following activities)

Activity Slack

S = LS - ES = LF - EF

SUMMARY OF KEY TERMS