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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

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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

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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.

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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.

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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.

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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 )).

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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.

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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

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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.

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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 -- 4 6 8 A -- 4 6 8 B -- 1 4.5 5 B -- 1 4.5 5 C A 3 3 3 C A 3 3 3 D A 4 5 6 D A 4 5 6 E A 0.5 1 1.5 E A 0.5 1 1.5 F B,C 3 4 5 F B,C 3 4 5 G B,C 1 1.5 5 G B,C 1 1.5 5 H E,F 5 6 7 H E,F 5 6 7 I E,F 2 5 8 I E,F 2 5 8 J D,H 2.5 2.75 4.5 J D,H 2.5 2.75 4.5 K G,I 3 5 7 K G,I 3 5 7

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11 Slide Example: ABC Associates n PERT Network Representation

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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

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13 Slide Example: ABC Associates n Earliest/Latest Times Activity ES EF LS LF Slack A 0 6 0 6 0 *critical A 0 6 0 6 0 *critical B 0 4 5 9 5 B 0 4 5 9 5 C 6 9 6 9 0 * C 6 9 6 9 0 * D 6 11 15 20 9 D 6 11 15 20 9 E 6 7 12 13 6 E 6 7 12 13 6 F 9 13 9 13 0 * F 9 13 9 13 0 * G 9 11 16 18 7 G 9 11 16 18 7 H 13 19 14 20 1 H 13 19 14 20 1 I 13 18 13 18 0 * I 13 18 13 18 0 * J 19 22 20 23 1 J 19 22 20 23 1 K 18 23 18 23 0 * K 18 23 18 23 0 * n Estimated Project Completion Time: Max EF = 23

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14 Slide Example: ABC Associates n Critical Path (A-C-F-I-K) 66 44 33 55 55 22 44 11 66 33 55 0 6 9 13 13 18 9 11 9 11 16 18 13 19 14 20 19 22 20 23 18 23 6 7 6 7 12 13 6 9 0 4 5 9 6 11 6 11 15 20

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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 + 0 + 1/9 + 1 + 4/9 = 4/9 + 0 + 1/9 + 1 + 4/9 = 2 = 2 = 1.414 = 1.414 z = (24 - 23)/ (24-23)/1.414 =.71 z = (24 - 23)/ (24-23)/1.414 =.71 From the Standard Normal Distribution table: From the Standard Normal Distribution table: P(z <.71) =.5 +.2612 =.7612 P(z <.71) =.5 +.2612 =.7612

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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.

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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.

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

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19 Slide Example: How Are We Doing? n Earliest/Latest Times Activity ES EF LS LF Slack Activity ES EF LS LF Slack A 0 9 0 9 0 A 0 9 0 9 0 B 0 8 5 13 5 B 0 8 5 13 5 C 0 10 7 17 7 C 0 10 7 17 7 D 8 11 22 25 14 D 8 11 22 25 14 E 8 12 13 17 5 E 8 12 13 17 5 F 9 13 13 17 4 F 9 13 13 17 4 G 9 12 9 12 0 G 9 12 9 12 0 H 12 17 12 17 0 H 12 17 12 17 0 I 12 16 21 25 9 I 12 16 21 25 9 J 17 25 17 25 0 J 17 25 17 25 0

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20 Slide Example: How Are We Doing? n Activity Status (end of eleventh week) Activity Actual Cost % Complete Activity Actual Cost % Complete A $6,200 100 A $6,200 100 B 5,700 100 B 5,700 100 C 5,600 90 C 5,600 90 D 0 0 D 0 0 E 1,000 25 E 1,000 25 F 5,000 75 F 5,000 75 G 2,000 50 G 2,000 50 H 0 0 H 0 0 I 0 0 I 0 0 J 0 0 J 0 0

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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 = 6000 - 300 B 5,700 (1.00)x6000 = 6000 - 300 C 5,600 (.90)x6000 = 5400 200 C 5,600 (.90)x6000 = 5400 200 D 0 0 0 D 0 0 0 E 1,000 (.25)x6000 = 1500 - 500 E 1,000 (.25)x6000 = 1500 - 500 F 5,000 (.75)x6000 = 4500 500 F 5,000 (.75)x6000 = 4500 500 G 2,000 (.50)x6000 = 3000 -1000 G 2,000 (.50)x6000 = 3000 -1000 H 0 0 0 H 0 0 0 I 0 0 0 I 0 0 0 J 0 0 0 J 0 0 0 Totals $25,500 $26,400 $- 900 Totals $25,500 $26,400 $- 900

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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.

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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).

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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.

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25 Slide The End of Chapter 12

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