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					      Basics Of PERT/CPM


PERT=Project Evaluation & Review Technique

        CPM = Critical Path Method
              Network

• Network is a graphic representation of
  a project's operations and is composed
  of activities and events that must be
  completed to reach the end of the
  objective of a project, showing the
  planning sequence of their
  accomplishment , their dependence and
  inter-relationships.
         Basic Component
• Activity is a task, or item of work to be
  done, that consume time, effort, money or
  other resources. Its lies between two events.
  An activity is presented by an arrow its
  head indicating the sequence in which the
  events are to occur
• Events is the start (beginning) or
  completion (end) of some activity and as
  such it consume no time, no resources. It is
  known as NODE.
             i             Activity
                                                j
       Starting Event                  Completion Event
Predecessor Activity: an activity which must be completed
before one or more other activities start is known as
predecessor activity(i predecessor j )

Successor activity: an activity which started immediately
after one or more of other activities are Completed is
known as successor activity (j successor i)

Dummy activity: an activity which does not consume
either any resource and time is known as dummy activity
(dotted line in the network)
         Local Sequencing

• The entire project may be considered
  as a series of activities which being
  only after another activity completed
• This types of relationship is called
  constrains which are represent by
  inequalities [eg. A<B (the activity A
  must be completed before the start of
  activity B)]
        Situations in network diagram
             B
    A                     A must finish before either B or C can start

                     C
    A
                 C        both A and B must finish before C can start
B
    A                C     both A and B must finish before either of C
    B                      or D can start
                     D

A        B
                         A must finish before B can start
        Dummy            both A and C must finish before D can start
C
                 D
    Errors in Network Diagram
• Looping : if an activity were represented as
  going back in time
                 B
         A

                 C



• Close loop produced endless cycle in
  computer programmes without a built-in
  routine for detection of identification of the
  cycle
Cycles
•A cycle is any path of jobs that leads back into itself.
•It represents logical error. It has to be removed before
scheduling computation.
•A job list with cycles cannot be put in topological
order.
       Errors in Network Diagram
• Dangling :No activity should end without being
  joined to maintain the continuity of the system.
  Such end – events other then the end of the
  projects as a whole are called dangling events


                         B           E
                 A


                         C           F
                     D
      Dangling

                             Dummy
          The CPM Diagram




• “Tasks” are Arrows    • “Events” are Circles
• “Critical Tasks” are Thick Arrows
• “Dummy Tasks” are Dashed Arrows
     Rules for Network Construction
1. Each activity is represented by one and only one arrow
2. Each activity must be identify by its starting and end node
–Two activities should not be identified by the same completion events
–Activity must be represented by their symbol or corresponding order pair of
starting and completion events
–Nodes are numbered to identify an activity uniquely.
3. Between any pair of nodes, there should be one and only one
activity
4. Arrow should be kept straight or not curved or bent.
5. The logical sequence between activities must follow following
rules
 - An event cannot occur until all the incoming activities into it have been
   completed
 - An activity cannot start unless all the preceding activities on which it
   depends have been completed
 - Dummy activities should only be introduced if obviously necessary
          Numbering the Events
• Event number should be unique.
• Event numbering should be carried out on a
  sequential basis from left to right.
• The initial event which has all outgoing arrows
  with no incoming arrows is numbered 0 or 1
• The head of an arrow should always bear a
  number higher than the one assigned at all tail of
  the arrow.
• Gaps should be left in the sequence of event
  numbering to accommodate subsequent inclusion
  of activities, if necessary.
                 Illustration - 1
• A television is manufactured in six steps, labeled A
  through F. Because of its size and complexity, the
  television is produced one at a time. The production
  control manager thinks that network scheduling techniques
  might be useful in planning future production. He recorded
  the following information
• A is the first step and precedes B and C
• C precedes D and E.
• B follows D and precedes E.
• D, E is successor of F.
        Activity on node diagram
•   A is the first step and precedes B and C (A < B, C)
•   C precedes D and E.
•   B follows D and precedes E.
•   D, E is successor of F.


                            B
               A
                             C
        Activity on node diagram
•   A is the first step and precedes B and C
•   C precedes D and E. (C < D, E)
•   B follows D and precedes E.
•   D, E is successor of F.


                             B                 D
               A
                             C                 E
        Activity on node diagram
•   A is the first step and precedes B and C
•   C precedes D and E.
•   B follows D and precedes E. (D < B > E)
•   D, E is successor of F.


                           B             D
               A
                            C             E
        Activity on node diagram
•   A is the first step and precedes B and C
•   C precedes D and E.
•   B follows D and precedes E.
•   D, E is successor of F. (F < D, E )


                             B                 D
               A                                   F
                             C                 E
                   Illustration - 2
Construct the network diagram comprising activities B, C,
…, Q and N such that the following constraints are
satisfies B < E, F; C < G, L; E, G < H; L, H < I; L < M; H
< N; H < J; I,J < P; P < Q.
                                   H           I
                   E       5           6               9
               2                                   J         10
                       F
           B               3                                      P
       1
               C           G                                          11
                       4                                     Q
                               L
                                       M           N
                                   7       8            12
                   Illustration - 3
Construct the network diagram comprising activities
A, B, C > NONE;
A<D
B, C < E
A<F
C<G
H < D, E, F
D<I
G < J, K
H, J < L
K<M
I, L < N
                  Illustration - 3
Construct the network diagram comprising activities
A, B, C > NONE;
A<D
B, C < E
A<F                            D
                           2       5      H
C<G
                                 F        I         9
H < D, E, F            A
                                       7
D<I                  1 C 4        G
G < J, K                               8
                          B
H, J < L
K<M                             E
                             3       6
I, L < N
                        Illustration - 4
Network analysis of a minor redesign of a product and its associated
packaging.
The key question is: How long will it take to complete this project ?
For clarity, this list is kept to a minimum by specifying only
immediate relationships, that is relationships involving activities
that "occur near to each other in time".
                      Practice Example
A social project manager is faced with a project with the following
activities:
 Activity Description                          Duration
 Social work team to live in village           5w
 Social research team to do survey             12w
 Analyse results of survey                     5w
 Establish mother & child health program       14w
 Establish rural credit programme              15w
 Carry out immunization of under fives         4w

 Draw network diagram and show the critical path.
 Calculate project duration.
                      Practice problem
Activity   Description                               Duration
1-2        Social work team to live in village       5w
1-3        Social research team to do survey         12w
3-4        Analyse results of survey                 5w
2-4        Establish mother & child health program   14w
3-5        Establish rural credit programme          15w
4-5        Carry out immunization of under fives     4w


                                   4
                  2

     1                                           5
                        3
                  CPM calculation
• Path
  – A connected sequence of activities leading from
    the starting event to the ending event
• Critical Path
  – The longest path (time); determines the project
    duration
• Critical Activities
  – All of the activities that make up the critical path
              Critical Path Analysis

• The objective of critical path analysis is to estimate
  the total project duration
  – Total duration needed for the completion of the project
  – The activities of the project as being critical or non-critical

 •An Activity in a network diagram is said to be
 critical is the delay in its start will further delay the
 project completion time.
        Critical Path Analysis


• The following terms shall be used in the
  critical path calculation

  – Ei = Earliest occurrence time of event i
  – Lj = Latest occurrence time of event j
  – Tij = duration of the activity (i, j)
Forward Pass
 • Earliest Start Time (ES)
    – earliest time an activity can start
    – ES = maximum EF of immediate predecessors
 • Earliest finish time (EF)
    – earliest time an activity can finish
    – earliest start time plus activity time
                                   EF= ES + t
Backward Pass
 Latest Start Time (LS)
    Latest time an activity can start without delaying critical path
    time
        LS= LF - t
 Latest finish time (LF)
    latest time an activity can be completed without delaying
    critical path time
    LS = minimum LS of immediate predecessors
CPM analysis
• Draw the CPM network
• Analyze the paths through the network
• Determine the float for each activity
   – Compute the activity’s float
                float = LS - ES = LF - EF
   – Float is the maximum amount of time that this activity can be
     delay in its completion before it becomes a critical activity,
     i.e., delays completion of the project
• Find the critical path is that the sequence of activities and events
  where there is no “slack” i.e.. Zero slack
   – Longest path through a network
• Find the project duration is minimum project completion time
                CPM Example:
• CPM Network
                      f, 15

                        g, 17             h, 9
       a, 6
                                i, 6

          b, 8
                       d, 13      j, 12

              c, 5
                      e, 9
                  CPM Example
• ES and EF Times        f, 15


                          g, 17             h, 9
         a, 6
        0 6                       i, 6

           b, 8
           0 8           d, 13      j, 12

                c, 5
           0 5          e, 9
                CPM Example
• ES and EF Times       f, 15
                        6 21
                         g, 17             h, 9
        a, 6
       0 6            6 23       i, 6

          b, 8
          0 8           d, 13      j, 12
                        8 21
               c, 5
          0 5          e, 9
                       5 14
                 CPM Example

• ES and EF Times         f, 15
                          6 21
                           g, 17            h, 9
          a, 6
                                           21 30
         0 6            6 23        i, 6
                                   23 29
            b, 8
            0 8           d, 13       j, 12
                          8 21          21 33
                 c, 5
            0 5          e, 9
                                     Project’s EF = 33
                         5 14
                  CPM Example
• LS and LF Times         f, 15
                          6 21
                                              h, 9
                                             21 30
         a, 6                g, 17
                                             24 33
        0 6           6 23            i, 6
                                     23 29
           b, 8                      27 33
           0 8          d, 13           j, 12
                        8 21
                                         21 33
             c, 5                        21 33
           0 5        e, 9
                       5 14
                  CPM Example
• LS and LF Times           f, 15
                            6 21
                                              h, 9
                          18 24
                                             21 30
        a, 6                 g, 17
                                             24 33
       0 6            6 23            i, 6
       4 10           10 27          23 29
           b, 8                      27 33
          0 8            d, 13          j, 12
          0 8            8 21            21 33
             c, 5        8 21            21 33
          0 5         e, 9
          7 12         5 14
                      12 21
               CPM Example
• Float
                              f, 15
                          3 6 21               h, 9
                              9 24
         a, 6                  g, 17        3 21 30
                                              24 33
      3 0 6           4 6 23              i, 6
        3 9             10 27
                                       4 23 29
            b, 8                         27 33
                          d, 13
        0 0 8                               j, 12
           0 8          0 8 21               0 21 33
               c, 5       8 21                  21 33
                           e, 9
          7 0 5
            7 12          7 5 14
                            12 21
                  CPM Example
• Critical Path         f, 15


                         g, 17             h, 9
         a, 6
                                 i, 6

            b, 8
                        d, 13      j, 12

                c, 5
                       e, 9
            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
• PERT is based on the assumption that an activity’s duration
  follows a probability distribution instead of being a single value
• Three time estimates are required to compute the parameters of
  an activity’s 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
      activity’s duration
   – optimistic time (to ) - the time the activity would take if things
      did go well

                                                 tp + 4 t m + to
       Mean (expected time):           te =
                                                         6
                                                             2
                                                  tp - to
                       Variance: Vt   =2   =
                                                     6
                  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-
          Z=
                  
 where  = tp = project mean time
          = 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

                            D


       A            E           H           J



                C

           B                        I   K
                        F

                            G
     PERT Example
Activity   Expected Time   Variance
     A           6          4/9
     B           4          4/9
     C           3            0
     D           5          1/9
     E           1          1/36
     F           4          1/9
     G           2          4/9
     H           6          1/9
     I           5            1
     J           3          1/9
     K           5          4/9
        PERT Example
Activity ES   EF   LS   LF   Slack
  A      0    6    0    6     0 *critical
  B      0    4    5    9     5
  C      6    9    6    9     0*
  D      6    11   15   20    9
  E      6    7    12   13    6
  F      9    13   9    13    0*
  G      9    11   16   18    7
  H     13    19   14   20    1
  I     13    18   13   18    0*
  J     19    22   20   23    1
  K     18    23   18   23    0*
        PERT Example

    Vpath = VA + VC + VF + VI + VK
          = 4/9 + 0 + 1/9 + 1 + 4/9
          = 2
    path = 1.414
   z = (24 - 23)/(24-23)/1.414 = .71
From the Standard Normal Distribution table:
     P(z < .71) = .5 + .2612 = .7612
   (a) Draw the activity network of the project
 (b) Find total float for each activity. Using above
information crash the activity step by step until all
                   path are critical.
     Activity       Normal time Crash time
        1-2               20               17
        1-3               25               25
        2-3               10                8
        2-4               12                6
        3-4                5                2
        4-5               10                5
        4-6                5                3
        5-7               10                5
        6-7                8                3
         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 . . haven’t 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.

				
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