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# Arrow computation

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• 1. CPM Network ComputationComputation Nomenclature• The following definitions and subsequent formulas will be given in terms of an arbitrary activity designed as (i-j) as shown below:Spring 2008, Arrow Diagramming 1King Saud University Dr. Khalid Al-Gahtani
• 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 ActivitiesSpring 2008, Arrow Diagramming 2King Saud University Dr. Khalid Al-Gahtani
• 3. Forward Pass ComputationsSTEP 1: E1 = 0STEP 2: Ei = Max all l (El + Dli) 2 ≤ i ≤ n.STEP 3: ESij = Ei all ij EFij = Ei + Dij all ijSTEP 4: The (Expected) project duration can be computed as the last activity (En) event time.Spring 2008, Arrow Diagramming 3King Saud University Dr. Khalid Al-Gahtani
• 4. Backward Pass ComputationsSTEP 1: Ln = Ts or EnSTEP 2: Lj = Minall k (Lk Djk) 1 ≤ j ≤ n-1STEP 3: LFij = Lj all ij LSij = Lj Dij all ijSpring 2008, Arrow Diagramming 4King Saud University Dr. Khalid Al-Gahtani
• 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 5Spring 2008, Arrow Diagramming 5King Saud University Dr. Khalid Al-Gahtani
• 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 25Spring 2008, Arrow Diagramming 6King Saud University Dr. Khalid Al-Gahtani
• 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 6Spring 2008, Arrow Diagramming 7King Saud University Dr. Khalid Al-Gahtani
• 8. Example 2:Spring 2008, Arrow Diagramming 8King Saud University Dr. Khalid Al-Gahtani
• 9. Forward pass calculationsStep 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 = 30Spring 2008, Arrow Diagramming 9King Saud University Dr. Khalid Al-Gahtani
• 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)Spring 2008, Arrow Diagramming 10King Saud University Dr. Khalid Al-Gahtani
• 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.Spring 2008, Arrow Diagramming 11King Saud University 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:Spring 2008, Arrow Diagramming 12King Saud University Dr. Khalid Al-Gahtani
• 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 – EFijSpring 2008, Arrow Diagramming 13King Saud University Dr. Khalid Al-Gahtani
• 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 Spring 2008, Arrow Diagramming 14 King Saud University 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) = Fij AFij = Lj EjSpring 2008, Arrow Diagramming 15King Saud University 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 (IDFij) = Max (0, Ej Li –Dij) = Max (0, Min (ESjk) - Max (LFli) Dij)Spring 2008, Arrow Diagramming 16King Saud University Dr. Khalid Al-Gahtani
• 17. E S ij E F ij E S jk L F ij AF IT F ID F FSpring 2008, Arrow Diagramming 17King Saud University 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, Ej Li –Dij) = Max (0, Min (ESjk) Max (LFli) Dij)Spring 2008, Arrow Diagramming 18King Saud University Dr. Khalid Al-Gahtani
• 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 3Spring 2008, Arrow Diagramming 19King Saud University Dr. Khalid Al-Gahtani
• 20. Example 3: 1 B A 0 3 C E X D F G 2 4 5 6Spring 2008, Arrow Diagramming 20King Saud University Dr. Khalid Al-Gahtani
• 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 32Spring 2008, Arrow Diagramming 21King Saud University Dr. Khalid Al-Gahtani
• 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.Spring 2008, Arrow Diagramming 22King Saud University 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 networkSpring 2008, Arrow Diagramming 23King Saud University 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.Spring 2008, Arrow Diagramming 24King Saud University Dr. Khalid Al-Gahtani
• 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-daySpring 2008, Arrow Diagramming 25King Saud University Dr. Khalid Al-Gahtani