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CPM Example

Critical Path
f, 15 g, 17 i, 6 b, 8 d, 13 c, 5 e, 9 j, 12 h, 9

a, 6

Critical Path Analysis

A critical path consists that set of dependent tasks (each dependent on the preceding one), which together take the longest time to complete. One way is to draw critical path tasks with a double line instead of a single line. The critical path for any given method may shift as the project progresses; this can happen when tasks are completed either behind or ahead of schedule, causing other tasks which may still be on schedule to fall on the new critical path

PERT is based on the assumption that an activitys duration follows a probability distribution instead of being a single value Three time estimates are required to compute the parameters of an activitys duration distribution: pessimistic time (tp ) - the time the activity would take if things did not go well most likely time (tm ) - the consensus best estimate of the activitys duration optimistic time (to ) - the time the activity would take if things did go well Mean (expected time): te =

tp + 4 tm + to

Variance: Vt =W2 =

tp - to

PERT analysis
Draw the network. Analyze the paths through the network and find the critical path. The length of the critical path is the mean of the project duration probability distribution which is assumed to be normal The standard deviation of the project duration probability distribution is computed by adding the variances of the critical activities (all of the activities that make up the critical path) and taking the square root of that sum Probability computations can now be made using the normal distribution table.

Probability computation
Determine probability that project is completed within specified time x-Q Z= W where Q = tp = project mean time W = project standard mean time x = (proposed ) specified time

PERT Example
Immed. Optimistic Most Likely Pessimistic Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.) A -4 6 8 B -1 4.5 5 C A 3 3 3 D A 4 5 6 E A 0.5 1 1.5 F B,C 3 4 5 G B,C 1 1.5 5 H E,F 5 6 7 I E,F 2 5 8 J D,H 2.5 2.75 4.5 K G,I 3 5 7

PERT Example PERT Network



PERT Example

Expected Time
6 4 3 5 1 4 2 6 5 3 5

4/9 4/9 0 1/9 1/36 1/9 4/9 1/9 1 1/9 4/9

PERT Example
Activity ES
A B C D E F G H I J K 0 0 6 6 6 9 9 13 13 19 18

6 4 9 11 7 13 11 19 18 22 23

0 5 6 15 12 9 16 14 13 20 18

6 9 9 20 13 13 18 20 18 23 23

0 *critical 5 0* 9 6 0* 7 1 0* 1 0*

PERT Example Vpath = VA + VC + VF + VI + VK = 4/9 + 0 + 1/9 + 1 + 4/9 = 2 Wpath = 1.414 z = (22 - 23)/W!(22-23)/1.414 = -0.71 From the Standard Normal Distribution table: P(z < 0.71) = .5 + .2612 = .7612

We will use PERT/CPM Analysis to determine Task Secondary properties:

Tail Event and Head Event Earliest Start, Earliest Complete Latest Start, Latest Complete Critical / Non-Critical Status Total Float, Free Float Scheduled Start, Scheduled Complete Actual Staffing, Duration, and Variable Costs

We will then use Task Secondary Properties to generate Project Management Tools:
Gantt Chart (Project Schedule) Manpower Chart Expenditure Curves Project Completion (PC)

Generate Initial CPM Diagram

Must strictly enforce all prerequisite relationships. Number of events is initially unknown Critical path is initially unknown Iterative Process Try to minimize number of Dummy Tasks

CPM Hint #1

Add or remove events at your pleasure. Do not number events until last.

CPM Hint #2
The initial event is the Tail Event for all tasks which have empty prerequisite sets (Initial Tasks). The Final Event is the Head Event for all tasks which are not members of any prerequisite set (Final Tasks).

CPM Hint #3
Tasks which have identical prerequisite sets have the same Tail Event

CPM Hint #4
Starting with the Final Tasks, work backwards, enforcing the smallest prerequisite sets first. Use Dummy Tasks to enforce any prerequisites in large sets which have already been enforced in a smaller set.

Finish CPM Diagram

Remove all redundant Dummy Tasks Remove all redundant Events Number all remaining events Not really finished . . havent identified critical tasks yet.

Generate PERT Chart: Enter Data for Each Task

Task Symbol Tail Event Head Event Task Duration (TD)

Forward Pass: Determine Earliest Start (ES) and Earliest Complete (EC) for each Task
For all Initial Tasks, ES = 0 Once ES is Determined, EC equals ES plus TD. The ES for all tasks with tail [i] is equal to the largest value of EC for all tasks with head [i]. PC is the largest value of EC for all Final Tasks.

Backward Pass: Determine Latest Start (LS) and Latest Complete (LC) for each Task
For all Final Tasks, LC = PC Once LC is Determined, LS equals LC minus TD. The LC for all tasks with head [j], is equal to the smallest value of LS for all tasks with tail [j]. At least one Initial Task must have LS = 0; none may be negative.

Determine Total Float (TF): Allowable delay in start of task which will not delay Project Completion
For task with tail [i] and head [j], TF[i,j] = (LC[j] ES[i]) TD[i,j] ES[i] is earliest start for all tasks with tail [i]. LC[j] is latest complete for all tasks with head [j].

Determine Free Float (FF): Allowable delay in start of task which will not delay start of any other task.
For task with tail [i] and head [j], FF[i,j] = ES[j] - ES[i] - TD[i, j] = ES[j] - EC[i,j] If [j] is the final event, use PC for ES[j]

Determine Critical Path

All Tasks with zero Total Float are Critical. Any delay in these Tasks will delay Project Completion. Darken these Tasks to finish CPM Diagram.