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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 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

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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 *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

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

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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 = 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

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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 A B B C C D D E E F F G G H H I I J J

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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, 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

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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 = 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

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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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