This document defines key terms and formulas used in critical path method (CPM) network computation. It discusses the forward and backward pass computations used to calculate the earliest and latest start and finish times of activities. The forward pass uses the earliest start time of predecessor activities to calculate the earliest finish time of each activity. The backward pass uses the latest finish time to calculate the latest start time. Several examples are provided to demonstrate applying CPM calculations to activity networks. Key outputs include identifying the critical path and activities that dictate the minimum project duration.

1. CPM Network Computation
Computation Nomenclature
• The following definitions and
subsequent formulas will be given in terms
of an arbitrary activity designed as (i-j) as
shown below:
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2. Computation Nomenclature
l k
Ei Ej
A C T (E S ij, E F ij)
i j
D ij (L S ij, L F ij)
Li Lj
k
l
Predecessors Successors
Activities Activities
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3. Forward Pass Computations
STEP 1: E1 = 0
STEP 2: Ei = Max all l (El + Dli) 2 ≤ i ≤ n.
STEP 3: ESij = Ei all ij
EFij = Ei + Dij all ij
STEP 4: The (Expected) project duration can be
computed as the last activity (En) event time.
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4. Backward Pass Computations
STEP 1: Ln = Ts or En
STEP 2: Lj = Minall k (Lk Djk) 1 ≤ j ≤ n-1
STEP 3: LFij = Lj all ij
LSij = Lj Dij all ij
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5. Example 1:
A ct ivit y ID D epend s o n T im e ES EF LS LF
A (1 -2) 5
B (2 -3) A 15
C (2 -4) A 10
D u m m y (3 -4)
D (3 -5) B 15
E (4-5) B, C 10
F (5 -6) D, E 5
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6. Example 1:
20
3
B 20 D
0 5 15 15 35 40
A F
1 2 5 6
5 5
0 5 C E 35 40
10 20 10
4
25
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7. Example 2:
A ct ivit y D escript io n P redecesso rs D uratio n
A S ite clearing --- 4
B R e m o va l o f trees --- 3
C G enera l e xca vat io n A 8
D G rad ing genera l area A 7
E E xca vat io n fo r trenche s B, C 9
F P lac ing fo rm w o rk and reinfo rce m e nt fo r co ncrete B, C 12
G Insta lling sew er line s D, E 2
H Insta lling other utilit ie s D, E 5
I P o uring co ncrete F, G 6
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8. Example 2:
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9. Forward pass calculations
Step 1  E0 = 0
Step 2
j= 1  E1 = M ax{E 0 + D 0 1 } = M ax{ 0 + 4 } = 4
j= 2  E2 = M ax{E 0 + D 0 2 ; E (1 ) + D 1 2 } = M ax{0 + 3; 4 + 8} = 12
j= 3  E3 = M ax{E 1 + D 1 3 ; E (2) + D 2 3 } = M ax{4 + 7; 12 + 9} = 21
j= 4  E4 = M ax{E 2 + D 2 4 ; E (3) + D 3 4 } = M ax{12 + 12; 21 + 2} = 24
j= 5  E5 = M ax{E 3 + D 3 5 ; E (4) + D 4 5 } = M ax{21 + 5; 24 + 6} = 30
the minimum time required to complete the project is 30 since E5 = 30
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10. Backward pass calculations
Step 1  L 5 = E 5 = 30
Step 2
j= 4  L4 = M in {L 5 - D 45 } = M in { 3 0 - 6} = 24
j= 3  L3 = M in {L 5 - D 35 ; L 4 - D 3 4 } = M in {30 -5; 24 - 2} = 22
j= 2  L2 = M in {L 4 - D 24 ; L 3 - D 2 3 } = M in {24 - 12; 22 - 9} = 12
j= 1  L1 = M in {L 3 - D 13 ; L 2 - D 1 2 } = M in {22 - 7; 12 - 8} = 4
j= 0  L0 = M in {L 2 - D 02 ; L 1 - D 0 1 } = M in {12 - 3; 4 - 4} = 0
• E0 = L0, E1 = L1, E2 = L2, E4 = L4,and E5 = L5.
• As a result, all nodes but node 3 are in the critical path.
• Activities on the critical path include:
A (0,1), C (1,2), F (2,4) and I (4,5)
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11. Final Results of Example 1
E arlie st E arlie st L atest L atest
D uratio n
A ct ivit y start tim e finish t im e start tim e finish t im e
D ij
E S ij = E i E F ij= E S ij + D ij L S ij= L F ij D ij L i = L F ij
A (0,1) 4 0* 4* 0 4*
B (0,2) 3 0 3 9 12
C (1,2) 8 4* 1 2* 4 12*
D (1,3) 7 4 11 15 22
E (2,3) 9 12 21 13 22
F (2,4) 12 12* 2 4* 12 24*
G (3,4) 2 21 23 22 24
H (3,5) 5 21 26 25 30
I (4,5) 6 24* 30* 24* 30*
*Activity on a critical path since Ei + Dij = Lj.
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12. Float and their Management
• Float Definitions:
– Float or Slack is the spare time available or
not critical activities.
– Indicates an amount of flexibility associated
with an activity.
– There are four various categories of activity
float:
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13. 1. Total Float:
• Total Float or Path Float is the maximum
amount of time that the activity can be delayed
without extending the completion time of the
project.
• It is the total float associated with a path.
• For arbitrary activity (i j), the Total Float can
be written as:
• Path Float Total Float (Fij) = LSij ESij
= LFij EFij
= Lj – EFij
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14. 2. Free Float
• Free Float or Activity Float is equal to the amount
of time that the activity completion time can be
delayed without affecting the earliest start or
occurrence time of any other activity or event in the
network.
• It is owned by an individual activity, whereas path
or total float is shared by all activities a long slack
path.
• can be written as:
Activity Float Free Float (AFij) = Min (ESjk) EFij
= Ej EFij
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15. 3. Interfering Float:
• That if used will effect the float of other
activities along its path (shared float).
• For arbitrary activity (i j), the Interfering
Float can be written as:
Interfering Float (ITFij) = Fij AFij
= Lj Ej
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16. 4. Independent Float
• It is the amount of float which an activity will
always possess no matter how early or late it
or its predecessors and successors are.
• Float that is “owned” by one activity.
• In all cases, independent float is always less
than or equal to free float.
• can be written as:
Independent Float (IDFij) = Max (0, Ej Li –Dij)
= Max (0, Min (ESjk) -
Max (LFli) Dij)
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17. E S ij E F ij E S jk L F ij
AF IT F
ID F
F
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18. Float Computations
Path Float Total Float (Fij) = LSij ESij
= LFij EFij
= Lj – EFij
Activity Float Free Float (AFij) = Min (ESjk) EFij
= Ej EFij
Interfering Float (ITFij) = Fij AFij
= Lj Ej
Independent Float (IDFij) = Max (0, Ej Li –Dij)
= Max (0, Min (ESjk)
Max (LFli) Dij)
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19. Example 3:
A ct ivit y D escript io n P redecesso rs D uratio n
A P re lim inary desig n --- 6
B E va luat io n o f desig n A 1
C C o ntract negotiat io n --- 8
D P reparatio n o f fa bricat io n p la nt C 5
E F ina l de sig n B, C 9
F Fa bricat io n o f P ro duct D, E 12
G S hip m e nt o f P ro duct to o w ner F 3
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20. Example 3:
1
B
A
0 3
C E
X
D F G
2 4 5 6
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21. Example 3:
E arlie st T im e L atest T im e
N o de
Ei Li
0 0 0
1 6 7
2 8 8
3 8 8
4 17 17
5 29 29
6 32 32
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22. Example 3:
E arlie st L atest T otal Free Interfering Indepe nde nt
A ct ivit y start tim e start tim e F lo at F lo at F lo at F lo at
E S ij L S ij F ij A F ij IT F ij ID F ij
A (0,1) 0 1 1 0 1 0
B (1,3) 6 7 1 1 0 0
C (0,2) 0 0 0 0 0 0
D (2,4) 8 12 4 4 0 4
E (3,4) 8 8 0 0 0 0
F (4,5) 17 17 0 0 0 0
G (5,6) 29 29 0 0 0 0
X (2,3) 8 8 0 0 0 0
• The minimum completion time for the project is 32 days
• Activities C,E,F,G and the dummy activity X are seen to lie on the critical path.
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23. Critical Path Identifications
• The critical path is continues chain of activities from the
beginning to the end, with zero float (if the zero-float
convention of letting Lt = Et for terminal network event is
followed).
• The critical path is the one with least path float (if the
zero-float convention of letting Lt = Et for terminal
network event is NOT followed).
• The longest path through the network.
• T = ∑ ti*, where
– T = project Completion Time
– ti* = Duration of Critical Activity
• There may be more than one critical paths in a network
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24. Identify CP activities & path(s)
1. Critical Activity:
• An activity for which no extra time is available
(no float, F = 0). Any delay in the completion of
a critical activity will delay the project duration.
2. Critical Path:
• Joins all the critical activities.
• Is the longest time path in the network?
• CP’s could be multiple in a project network.
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25. Ownership of float
Allow
Float Allow Prevent Ability to
Flexibility Prevent Solve TF
Float Ownership Flexibility disentitled Distribute TF
to include Schedule changing
Ownership issues for Resource float among project
change Games issues
concepts leveling consumption parties
order
Contractor ✓ ✕ ✕ ✓ ✕ ✕
Owner ✕ ✓ ✕ ✕ ✕ ✕
Project # # * * ✕ ✕
Bar1 ✕ ✕ ✓ ✕ ✕ ✕
50/502 # # * * ✕ ✕
Contract Risk3 ✓ ✕ ✓ ✓ ✕ ✕
Path Distribution4 ✓ ✕ ✓ ✓ ✓ ✕
Commodity5 ✓ ✓ * ✓ ✕ ✕
Day-by-day ✕ ✕ ✕ ✕ ✕ ✓
Contract Risk +
Path Distribution +
✓ ✓ ✓ ✓ ✓ ✓
Commodity +
Day-by-day
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