9-1

Chapter 9Chapter 9

Project Scheduling Project Scheduling

McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

9-2

The Elements of Project SchedulingThe Elements of Project Scheduling

Project DefinitionProject Definition. Statement of project, goals, . Statement of project, goals, and resources required.and resources required.

Activity DefinitionsActivity Definitions. Content and requirements of . Content and requirements of each activity.each activity.

Project SchedulingProject Scheduling. Specification of starting and . Specification of starting and ending times of all activities.ending times of all activities.

Project MonitoringProject Monitoring. Keeping track of the progress . Keeping track of the progress of the project. of the project.

9-3

Network Representation Network Representation

Projects may be represented as networks with:Projects may be represented as networks with:

Arrows representing activities.Arrows representing activities.

Nodes representing completion of a set of Nodes representing completion of a set of activities (milestones). activities (milestones).

Pseudo activities may be required to satisfy Pseudo activities may be required to satisfy precedence relationships. precedence relationships.

(Figure 9-4 (next) shows a typical project network.)(Figure 9-4 (next) shows a typical project network.)

9-4Correct Network Correct Network Representation for Example 9.3Representation for Example 9.3

9-5

Critical Path MethodCritical Path Method

An analytical tool that provides a schedule that An analytical tool that provides a schedule that completes the project in minimum time subject to the completes the project in minimum time subject to the precedence constraints. In addition, CPM provides:precedence constraints. In addition, CPM provides:

Starting ending times for each activityStarting ending times for each activity Identification of the critical activities (i.e., the ones Identification of the critical activities (i.e., the ones

whose delay necessarily delay the project). whose delay necessarily delay the project). Identification of the non-critical activities, and the Identification of the non-critical activities, and the

amount of slack time available when scheduling amount of slack time available when scheduling these activities. these activities.

9-6

Time Costing MethodsTime Costing Methods

Suppose that projects can be expedited by reducing the Suppose that projects can be expedited by reducing the time required for critical activities. Doing so results in time required for critical activities. Doing so results in an increase in some costs and a decrease in others. The an increase in some costs and a decrease in others. The goal is to determine the optimal number of days to goal is to determine the optimal number of days to schedule the project to minimize total cost. schedule the project to minimize total cost.

Assume that there is a linear time/cost relationship for Assume that there is a linear time/cost relationship for each activity. (See Figure 9-10). each activity. (See Figure 9-10).

Since direct costs decline with the project time and Since direct costs decline with the project time and indirect costs increase with the project time, the total indirect costs increase with the project time, the total cost curve is a convex function whose minimum cost curve is a convex function whose minimum corresponds to the optimal solution (See Figure 9-11). corresponds to the optimal solution (See Figure 9-11).

9-7The CPM Cost-Time The CPM Cost-Time Linear ModelLinear Model

9-8Optimal Project Optimal Project Completion TimeCompletion Time

9-9

PERT: Project Evaluation and Review PERT: Project Evaluation and Review TechniqueTechnique

PERT is a generalization of CPM to allow for PERT is a generalization of CPM to allow for uncertain activity times. For each activity the user uncertain activity times. For each activity the user must specify:must specify:a = minimum completion timea = minimum completion timeb = maximum completion timeb = maximum completion timem = most likely completion timem = most likely completion time

The method assumes each activity time follows a beta The method assumes each activity time follows a beta distribution, which can be fit precisely with distribution, which can be fit precisely with specification of a, b, and m. specification of a, b, and m.

(See Figure 9-12 for an example with a= 5, b=20 and (See Figure 9-12 for an example with a= 5, b=20 and m=17). m=17).

9-10Probability Density Probability Density of Activity Timeof Activity Time

9-11

PERT (continued)PERT (continued)

The mean and standard deviation of activity times are The mean and standard deviation of activity times are estimated from the following formulas (based on the estimated from the following formulas (based on the beta distribution)beta distribution)

In PERT one assumes that the path the with longest In PERT one assumes that the path the with longest expected completion time is the true critical path (this is expected completion time is the true critical path (this is only an approximation, since true critical path is a only an approximation, since true critical path is a random variable). random variable).

4 and

6 6

a m b b a

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PERT (concluded)PERT (concluded)

One assumes that the expected value of the project One assumes that the expected value of the project completion time is the sum of the expected values of completion time is the sum of the expected values of the critical activities and variance of the project the critical activities and variance of the project completion time is the sum of the variances of the completion time is the sum of the variances of the critical activities. (This is strictly true if the activity critical activities. (This is strictly true if the activity times are independent random variables.)times are independent random variables.)

Finally, one invokes the Central Limit Theorem to Finally, one invokes the Central Limit Theorem to conclude that the total project completion time is a conclude that the total project completion time is a random variable whose distribution is approximately random variable whose distribution is approximately normal. normal.

9-13

Resource ConsiderationsResource Considerations

When multiple projects compete for resources (such as When multiple projects compete for resources (such as materials and worker time), projects schedules may be materials and worker time), projects schedules may be impacted due to insufficient resources.impacted due to insufficient resources.

For example, consider two projects requiring Resources A For example, consider two projects requiring Resources A

and B as pictured Figure 9-20.and B as pictured Figure 9-20.

One can generate a resource load profile such as the one in One can generate a resource load profile such as the one in

Figure 9-21 to be certain that critical resources are sufficient Figure 9-21 to be certain that critical resources are sufficient to meet project requirements. to meet project requirements.

9-14Two Projects Two Projects Sharing Two ResourcesSharing Two Resources

9-15

Load Profiles for RAM and Permanent Load Profiles for RAM and Permanent Memory (Refer to Example 9.10)Memory (Refer to Example 9.10)