9-1 Chapter 9 Project Scheduling Chapter 9 Project Scheduling McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

9-2 The Elements of Project Scheduling n Project Definition. Statement of project, goals, and resources required. n Activity Definitions. Content and requirements of each activity. n Project Scheduling. Specification of starting and ending times of all activities. n Project Monitoring. Keeping track of the progress of the project.

9-3 Network Representation Projects may be represented as networks with: n Arrows representing activities. n Nodes representing completion of a set of activities (milestones). n Pseudo activities may be required to satisfy precedence relationships. (Figure 9-4 (next) shows a typical project network.)

9-4 Correct Network Representation for Example 9.3

9-5 Critical Path Method An analytical tool that provides a schedule that completes the project in minimum time subject to the precedence constraints. In addition, CPM provides: n Starting ending times for each activity n Identification of the critical activities (i.e., the ones whose delay necessarily delay the project). n Identification of the non-critical activities, and the amount of slack time available when scheduling these activities.

9-6 Time Costing Methods n Suppose that projects can be expedited by reducing the time required for critical activities. Doing so results in an increase in some costs and a decrease in others. The goal is to determine the optimal number of days to schedule the project to minimize total cost. n Assume that there is a linear time/cost relationship for each activity. (See Figure 9-10). n Since direct costs decline with the project time and indirect costs increase with the project time, the total cost curve is a convex function whose minimum corresponds to the optimal solution (See Figure 9-11).

9-7 The CPM Cost-Time Linear Model

9-8 Optimal Project Completion Time

9-9 PERT: Project Evaluation and Review Technique PERT is a generalization of CPM to allow for uncertain activity times. For each activity the user must specify: a = minimum completion time b = maximum completion time m = most likely completion time The method assumes each activity time follows a beta distribution, which can be fit precisely with specification of a, b, and m. (See Figure 9-12 for an example with a= 5, b=20 and m=17).

9-10 Probability Density of Activity Time

9-11 PERT (continued) The mean and standard deviation of activity times are estimated from the following formulas (based on the beta distribution) In PERT one assumes that the path the with longest expected completion time is the true critical path (this is only an approximation, since true critical path is a random variable).

9-12 PERT (concluded) One assumes that the expected value of the project completion time is the sum of the expected values of the critical activities and variance of the project completion time is the sum of the variances of the critical activities. (This is strictly true if the activity times are independent random variables.) Finally, one invokes the Central Limit Theorem to conclude that the total project completion time is a random variable whose distribution is approximately normal.

9-13 Resource Considerations n When multiple projects compete for resources (such as materials and worker time), projects schedules may be impacted due to insufficient resources. n For example, consider two projects requiring Resources A and B as pictured Figure n One can generate a resource load profile such as the one in Figure 9-21 to be certain that critical resources are sufficient to meet project requirements.

9-14 Two Projects Sharing Two Resources

9-15 Load Profiles for RAM and Permanent Memory (Refer to Example 9.10)