1. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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:

2. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 2 Computation Nomenclature Predecessors Activities Successors Activities

3. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 3 Forward Pass Computations STEP 1: E 1 = 0 STEP 2: E i = Max all l (E l + D li )2 ≤ i ≤ n. STEP 3: ES ij = E i all ij EF ij = E i + D ij all ij STEP 4:The (Expected) project duration can be computed as the last activity (E n ) event time.

4. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 4 Backward Pass Computations STEP 1: L n = T s or E n STEP 2: L j = Min all k (L k  D jk ) 1 ≤ j ≤ n-1 STEP 3: LF ij = L j all ij LS ij = L j  D ij all ij

5. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 5 Example 1:

6. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 6 Example 1:

7. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 7 Example 2:

8. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 8 Example 2:

9. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 9 Forward pass calculations  the minimum time required to complete the project is 30 since E5 = 30

10. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 10 Backward pass calculations 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)

11. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 11 Final Results of Example 1 *Activity on a critical path since E i + D ij = L j.

12. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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:

13. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 13 1. Total Float: Total FloatPath FloatTotal 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 (F ij )= LS ij  ES ij = LF ij  EF ij = L j – EF ij

14. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 14 2. Free Float Free FloatActivity FloatFree 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 (AF ij )= Min (ES jk )  EF ij = E j  EF ij

15. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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) = F ij  AF ij = L j  E j

16. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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 (IDF ij )= Max (0, E j  L i –D ij ) = Max (0, Min (ES jk ) - Max (LF li )  D ij )

17. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 17

18. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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, E j  L i –D ij ) = Max (0, Min (ESjk)  Max (LFli)  Dij)

19. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 19 Example 3:

20. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 20 Example 3:

21. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 21 Example 3:

22. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 22 Example 3: 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.

23. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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

24. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 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.

25. Spring 2008, King Saud University Arrow Diagramming Dr. Khalid Al-Gahtani 25 Ownership of float