1 1 Slide © 2001 South-Western College Publishing/Thomson Learning Anderson Sweeney Williams Anderson Sweeney Williams Slides Prepared by JOHN LOUCKS QUANTITATIVE METHODS FOR BUSINESS 8e QUANTITATIVE METHODS FOR BUSINESS 8e

2 2 Slide Chapter 12 Project Scheduling: PERT/CPM n Project Scheduling with Known Activity Times n Project Scheduling with Uncertain Activity Times n Considering Time-Cost Trade-Offs

3 3 Slide PERT/CPM n PERT stands for Program Evaluation Review Technique. n CPM stands for Critical Path Method. n PERT/CPM is used to plan the scheduling of individual activities that make up a project. n PERT/CPM can be used to determine the earliest/latest start and finish times for each activity, the entire project completion time and the slack time for each activity.

4 4 Slide Project Network n A project network can be constructed to model the precedence of the activities. n The nodes of the network represent the activities. n The arcs of the network reflect the precedence relationships of the activities. n A critical path for the network is a path consisting of activities with zero slack.

5 5 Slide Determining the Critical Path n Step 1: Make a forward pass through the network as follows: For each activity i beginning at the Start node, compute: Earliest Start Time = the maximum of the earliest finish times of all activities immediately preceding activity i. (This is 0 for an activity with no predecessors.)Earliest Start Time = the maximum of the earliest finish times of all activities immediately preceding activity i. (This is 0 for an activity with no predecessors.) Earliest Finish Time = (Earliest Start Time) + (Time to complete activity i.Earliest Finish Time = (Earliest Start Time) + (Time to complete activity i. The project completion time is the maximum of the Earliest Finish Times at the Finish node.

6 6 Slide Determining the Critical Path n Step 2: Make a backwards pass through the network as follows: Move sequentially backwards from the Finish node to the Start node. At a given node, j, consider all activities ending at node j. For each of these activities, ( i, j ), compute: Latest Finish Time = the minimum of the latest start times beginning at node j. (For node N, this is the project completion time.)Latest Finish Time = the minimum of the latest start times beginning at node j. (For node N, this is the project completion time.) Latest Start Time = (Latest Finish Time) - (Time to complete activity ( i, j )).Latest Start Time = (Latest Finish Time) - (Time to complete activity ( i, j )).

7 7 Slide Determining the Critical Path n Step 3: Calculate the slack time for each activity by: Slack = (Latest Start) - (Earliest Start), or Slack = (Latest Start) - (Earliest Start), or = (Latest Finish) - (Earliest Finish). = (Latest Finish) - (Earliest Finish). A critical path is a path of activities, from the Start node to the Finish node, with 0 slack times.

8 8 Slide Uncertain Activity Times n In the three-time estimate approach, the time to complete an activity is assumed to follow a Beta distribution. n An activity’s mean completion time is: t = ( a + 4 m + b )/6 t = ( a + 4 m + b )/6 n An activity’s completion time variance is:  2 = (( b - a )/6) 2  2 = (( b - a )/6) 2 a = the optimistic completion time estimate a = the optimistic completion time estimate b = the pessimistic completion time estimate b = the pessimistic completion time estimate m = the most likely completion time estimate m = the most likely completion time estimate

9 9 Slide Uncertain Activity Times n In the three-time estimate approach, the critical path is determined as if the mean times for the activities were fixed times. n The overall project completion time is assumed to have a normal distribution with mean equal to the sum of the means along the critical path and variance equal to the sum of the variances along the critical path.

10 Slide Example: ABC Associates n Consider the following project: Immed. Optimistic Most Likely Pessimistic Immed. Optimistic Most Likely Pessimistic Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.) A A B B C A C A D A D A E A E A F B,C F B,C G B,C G B,C H E,F H E,F I E,F I E,F J D,H J D,H K G,I K G,I 3 5 7

11 Slide Example: ABC Associates n PERT Network Representation

12 Slide Example: ABC Associates n Activity Expected Time and Variances t = ( a + 4 m + b )/6  2 = (( b - a )/6) 2 t = ( a + 4 m + b )/6  2 = (( b - a )/6) 2 Activity Expected Time Variance A 6 4/9 A 6 4/9 B 4 4/9 B 4 4/9 C 3 0 C 3 0 D 5 1/9 D 5 1/9 E 1 1/36 E 1 1/36 F 4 1/9 F 4 1/9 G 2 4/9 G 2 4/9 H 6 1/9 H 6 1/9 I 5 1 I 5 1 J 3 1/9 J 3 1/9 K 5 4/9 K 5 4/9

13 Slide Example: ABC Associates n Earliest/Latest Times Activity ES EF LS LF Slack A *critical A *critical B B C * C * D D E E F * F * G G H H I * I * J J K * K * n Estimated Project Completion Time: Max EF = 23

14 Slide Example: ABC Associates n Critical Path (A-C-F-I-K)

15 Slide Example: ABC Associates n Probability the project will be completed within 24 hrs  2 =  2 A +  2 C +  2 F +  2 H +  2 K = 4/ / /9 = 4/ / /9 = 2 = 2  =  = z = ( )/  (24-23)/1.414 =.71 z = ( )/  (24-23)/1.414 =.71 From the Standard Normal Distribution table: From the Standard Normal Distribution table: P(z <.71) = =.7612 P(z <.71) = =.7612

16 Slide PERT/Cost n PERT/Cost is a technique for monitoring costs during a project. n Work packages (groups of related activities) with estimated budgets and completion times are evaluated. n A cost status report may be calculated by determining the cost overrun or underrun for each work package. n Cost overrun or underrun is calculated by subtracting the budgeted cost from the actual cost of the work package. n For work in progress, overrun or underrun may be determined by subtracting the prorated budget cost from the actual cost to date.

17 Slide PERT/Cost n The overall project cost overrun or underrun at a particular time during a project is determined by summing the individual cost overruns and underruns to date of the work packages.

18 Slide Example: How Are We Doing? n Consider the following PERT network:

19 Slide Example: How Are We Doing? n Earliest/Latest Times Activity ES EF LS LF Slack Activity ES EF LS LF Slack A A B B C C D D E E F F G G H H I I J J

20 Slide Example: How Are We Doing? n Activity Status (end of eleventh week) Activity Actual Cost % Complete Activity Actual Cost % Complete A $6, A $6, B 5, B 5, C 5, C 5, D 0 0 D 0 0 E 1, E 1, F 5, F 5, G 2, G 2, H 0 0 H 0 0 I 0 0 I 0 0 J 0 0 J 0 0

21 Slide Example: How Are We Doing? n Cost Status Report (Assuming a budgeted cost of $6000 for each activity) (Assuming a budgeted cost of $6000 for each activity) Activity Actual Cost Value Difference A $6,200 (1.00)x6000 = 6000 $200 A $6,200 (1.00)x6000 = 6000 $200 B 5,700 (1.00)x6000 = B 5,700 (1.00)x6000 = C 5,600 (.90)x6000 = C 5,600 (.90)x6000 = D D E 1,000 (.25)x6000 = E 1,000 (.25)x6000 = F 5,000 (.75)x6000 = F 5,000 (.75)x6000 = G 2,000 (.50)x6000 = G 2,000 (.50)x6000 = H H I I J J Totals $25,500 $26,400 $- 900 Totals $25,500 $26,400 $- 900

22 Slide Example: How Are We Doing? n PERT Diagram at End of Week 11 The activity completion times are the times remaining for each activity.

23 Slide Example: How Are We Doing? n Corrective Action Note that the project is currently experiencing a $900 cost underrun, but the overall completion time is now 25.5 weeks or a.5 week delay. Management should consider using some of the $900 cost savings and apply it to activity G to assist in a more rapid completion of this activity (and hence the entire project). Note that the project is currently experiencing a $900 cost underrun, but the overall completion time is now 25.5 weeks or a.5 week delay. Management should consider using some of the $900 cost savings and apply it to activity G to assist in a more rapid completion of this activity (and hence the entire project).

24 Slide Critical Path Method n In the Critical Path Method (CPM) approach to project scheduling, it is assumed that the normal time to complete an activity, t j, which can be met at a normal cost, c j, can be crashed to a reduced time, t j ’, under maximum crashing for an increased cost, c j ’. n Using CPM, activity j 's maximum time reduction, M j, may be calculated by: M j = t j - t j '. It is assumed that its cost per unit reduction, K j, is linear and can be calculated by: K j = ( c j ' - c j )/ M j.

25 Slide The End of Chapter 12