# CEE 536 CRITICAL PATH METHODS EXAMPLE PROBLEMS. Photios G. Ioannou, PhD, PE Professor of Civil and Environmental Engineering

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1 CEE 536 CRITICAL PATH METHODS EXAMPLE PROBLEMS Photios G. Ioannou, PhD, PE Professor of Civil and Environmental Engineering Chachrist Srisuwanrat, Ph.D. Former Graduate Student Instructor University of Michigan Ann Arbor, Michigan

2 TABLE OF CONTENTS 1. NETWORK CONSTRUCTION ACTIVITY ON ARROW... 3 ACTIVITY ON NODE ACTIVITY-ON-ARROW SCHEDULING EVENT APPROACH MISSING FLOATS ACTIVITY-ON-NODE SCHEDULING SCHEDULING USING A LINK MATRIX PROJECT UPDATING PERT PNET TIME-COST TRADEOFF TABLES TIME-COST TRADEOFF-LP RESOUCE LEVELING MINIMUM MOMENT METHOD RESOUCE LEVELING PACK OVERLAPPING NETWORKS REPETITIVE SCHEDULING METHOD

3 1. NETWORK CONSTRUCTION ACTIVITY ON ARROW ACTIVITY ON NODE

4 Problem 1.1 a) Construct an activity on arrow network based on the activity descriptions below. Show all your work. Label activities in the network by their activity letters and node numbers. Remove any redundant dependencies and label dummy activities DUMMY1, DUMMY, etc. b) Construct a precedence network based on the same activity descriptions below. Show all your work. Label activities in the network by their activity letters and node numbers. Remove all redundant dependencies and arrange activities in proper sequence steps. Activities H, R, T1 start the project. Activity T can start when Activities H, E1 and S are completed. Activity E1 also depends on Activity R. Activity X follows Activity H and precedes Activity L. Activity E is preceded by Activities T and P1. The predecessors to Activity G are Activities L, T and P1. The successors to Activity T1 are Activities E1, S, W and D. Activity P1 cannot begin until Activity W is finished. Activity P and F follow Activities W and D, and precede Activities E and R1. Activity O depends on T and P1, and precedes Activity L. CEE536 Example Problems 4 P.G. Ioannou & C. Srisuwanrat

5 Problem 1. a) Construct an activity on arrow network based on the activity descriptions below. Show all your work. Label activities in the network by their activity letters and node numbers. Remove any redundant dependencies and label dummy activities DUMMY1, DUMMY, etc. b) Construct a precedence network based on the same activity descriptions below. Show all your work. Label activities in the network by their activity letters and node numbers. Remove all redundant dependencies and arrange activities in proper sequence steps. Activity I follows Activity B and precedes Activity Q. Activity B1 precedes Activity P and follows the completion of Activities Q, K, and E. Activity R follows the completion of Activity B. Activity S follows Activities R and S1, and precedes Activity P. Activity K3 is preceded by Activities X, L, and Z, and followed by Activities G and F. Activity E precedes Activities A1, X, L, and Z. Activity B can start when Activities A1 and X are completed. The predecessors to Activity S1 are Activities E, G, and F. Activity E depends on Activity L and E and precedes Activities N, S1, and K. Activity K follows Activities N, R, and L. Activity P depends on Activities R and N. Activity S depends on Activities X, F, and E. CEE536 Example Problems 5 P.G. Ioannou & C. Srisuwanrat

6 Problem 1.3 a) Construct an activity on arrow network based on the activity descriptions below. Show all your work. Label activities in the network by their activity letters and node numbers. Remove any redundant dependencies and label dummy activities DUMMY1, DUMMY, etc. b) Construct a precedence network based on the same activity descriptions below. Show all your work. Label activities in the network by their activity letters and node numbers. Remove all redundant dependencies and arrange activities in proper sequence steps. The predecessors to Activity Z are Activities L, C and R. The successors to Activity B are Activities E1, S, W and D. Activity E1 also depends on Activity M. Activity U and F follow Activities W and D, and precede Activities E and R1. Activity Y follows Activities C and R, and followed by Activity L. Activities D, M, and B start the project. Activity C can start when Activities D, E1 and S are completed. Activity R cannot begin until Activity W is finished. Activity I follows Activity D and precedes Activity L. Activity E follows Activities C and R. CEE536 Example Problems 6 P.G. Ioannou & C. Srisuwanrat

7 Solution 1.1.a CEE536 Example Problems 7 P.G. Ioannou & C. Srisuwanrat

8 Solution 1.1.b H X L G ST R E1 T O FIN T1 S P1 W P E D F R1 CEE536 Example Problems 8 P.G. Ioannou & C. Srisuwanrat

9 Solution 1..a B1 DUMMY 7 S1 K Z F L CEE536 Example Problems 9 P.G. Ioannou & C. Srisuwanrat

10 Solution 1..b CEE536 Example Problems 10 P.G. Ioannou & C. Srisuwanrat

11 D Solution 1.3.a DUMMY 3 U R1 W R DUMMY 4 DUMMY 5 E B S C Y Z M E1 L D DUMMY 1 I CEE536 Example Problems 11 P.G. Ioannou & C. Srisuwanrat

12 Solution 1.3.b D I L Z ST M E1 C Y FIN B S R W U E D F R1 CEE536 Example Problems 1 P.G. Ioannou & C. Srisuwanrat

13 . ACTIVITY-ON-ARROW SCHEDULING EVENT APPROACH MISSING FLOATS

14 Calculation for activity-on-arrow networks Note: TF => FF>= INDF and TF >= INTF. If TF = 0 then all the floats = 0. If FF = 0 then INDF = 0. Remember that INDF is a part of FF. CEE536 Example Problems 14 P.G. Ioannou & C. Srisuwanra

15 Problem.1 Calculate the schedule dates (TE and TL) and the four floats (TF, FF, INTF, and INDF). CEE536 Example Problems 15 P.G. Ioannou & C. Srisuwanra

16 Problem. Calculate the schedule dates (TE and TL) and the four floats (TF, FF, INTF, and INDF). CEE536 Example Problems 16 P.G. Ioannou & C. Srisuwanra

17 Problem.3 Calculate the schedule dates (TE and TL) and the four floats (TF, FF, INTF, and INDF). CEE536 Example Problems 17 P.G. Ioannou & C. Srisuwanra

18 Problem.4 Calculate the schedule dates (TE and TL) and the four floats (TF, FF, INTF, and INDF). CEE536 Example Problems 18 P.G. Ioannou & C. Srisuwanra

19 Solution.1 CEE536 Example Problems 19 P.G. Ioannou & C. Srisuwanra

20 Solution. CEE536 Example Problems 0 P.G. Ioannou & C. Srisuwanra

21 Solution.3 CEE536 Example Problems 1 P.G. Ioannou & C. Srisuwanra

22 Solution.4 CEE536 Example Problems P.G. Ioannou & C. Srisuwanra

23 MISSING FLOATS Calculate TF, FF, and INTF G H K C E R 3. TL = 3 T = 5 TF = 8 FF = CEE536 Example Problems 3 P.G. Ioannou & C. Srisuwanrat

24 4. 19 P 5 L T G H K M U 9 N R 1 S 7 5. TF =1 FF =4 INTF =3 FF =5 CEE536 Example Problems 4 P.G. Ioannou & C. Srisuwanrat

25 FOUR MISSING FLOAT CONCEPTS (please check these concepts with the previous activity-on-arrow practice) MS1. ZERO FREE FLOAT If there is only one link goes into a node, its FF = 0. FF of activity C = 0 If there are many links go into the same node, at least one of them must have FF = 0. FF of activity X = 0 MS. SAME INTERFERE FLOAT All the links that go into the same node have the same INTF. According to MS1, FF of X = 0. Thus, INTF of X = 5. According to MS, INTF of Z equals to INTF of X, which is 5. Thus, TF of Z is 11. CEE536 Example Problems 5 P.G. Ioannou & C. Srisuwanrat

26 MS3. ACTIVITY CHAIN S TOTAL FLOAT Total floats of activities on an activity chain are the same. Activity Q, W, and E are activity chain. Thus, TF of Q and E equal to TF of W according to MS3. FF of Q, W, and E equal to 0 according to MS1. CEE536 Example Problems 6 P.G. Ioannou & C. Srisuwanrat

27 Solution From MS1, at node 17, since two links go into the same node and FF of K = 3, FF of S = 0. Thus, INTF of S = FF = 0 INTF = 7 G H K From MS, links go into the same node have the same INTF. Thus, INTF of K = 7, and TF of K = FF = 0 INTF = 7 G H K INTF = 7 TF = From MS3, activities in the activity chain have the same TF. Thus, TF of K, G, and H are From MS1, FF of G and H = 0. INTF of G and H are FF = 0 INTF = 7 G H K TF = 10 FF = 0 INTF = 10 TF = 10 FF = 0 INTF = 10 INTF = 7 TF = 10 CEE536 Example Problems 7 P.G. Ioannou & C. Srisuwanrat

28 ..1) MS1 (one link goes into one node, FF of the link = 0) FF of A, B, C, and E= 0. So, INTF of A and E = 0 and 4 respectively. MS1 ( many links going to the same node, one of them must have zero FF ) FF of P = 0. Thus, INTF of P = 0. And also FF of R. 15 FF = 0 P FF = 0 INTF = 0 0 A 5 B 10 D FF = 0 FF = 0 INTF = 0 FF = 0 INTF = 4 0 U 5 30 FF = 0.) MS ( many links going into the same have the same INTF) INTF of U = INTF of P = 0. So, TF of U =. 15 FF = 0 P FF = 0 INTF = 0 0 A 5 B 10 D FF = 0 FF = 0 INTF = 0 FF = 0 INTF = 4 0 U 5 30 FF R = 0 INTF = 0 TF =.3) MS3 (activities on activity chain have the same TF) TF of B = TF of A. TF of R = TF of E. Then calculate INTF of R, which will give us INTF of D according MS. Thus, TF of D can be calculated. TF of C = TF of P. Note: up to this point, you should be able to get all the TF, FF, and also INTF. It should also be mentioned that the given TF of A and P are not necessary. WHY??? CEE536 Example Problems 8 P.G. Ioannou & C. Srisuwanrat

29 3. One link goes into one node FF = 0 Or Many links go into the same node at least one of the links must have FF = 0 TL = 7 from TLj-Tij TE = 1 form TLi-INTFhi TL = 3 TE = 3 INTFij = 6 T = 5 TF = 8 FF = INTFhi = 6 FF = 0! TF = = 6 INTF = ) Only one link goes into one node, we should be able to spot out activities whose FF = 0, which are activities G, H, N, L, P, and S. 4.) Many links go into the same node, at least one of them must have FF = 0. Thus, FF of T = 0 CEE536 Example Problems 9 P.G. Ioannou & C. Srisuwanrat

30 4.3) Total float on activities on an activity chain have the same TF Thus, we should be able to get TF of H and K (from Chain G-H-K), L and P (from Chain L-P-T). FF = 0 TF = 4 INTF = 4 19 FF = 0 TF = 4 INTF = 4 P 5 FF = 0 TF = 4 11 G 13 H 15 K 17 M FF = 0 FF K = 3 FF = 0 TF = 7 TF = 7 INTF = 7 INTF = 7 INTF = 4 U 3 9 FF = 0 INTF = 4 S 1 7 FF = 0 TF = 9 4.4) Activities going to the same node have the same INTF Thus, we can calculate INTF of M and R (going to node 3), and U (going to node 9). Note: INTF of M and R is TL 3 TE 3 (you should remember this by now) CEE536 Example Problems 30 P.G. Ioannou & C. Srisuwanrat

31 5. It is crucial to identify critical path, and knowing that any links going to the same node have the same INTF. TF =1 FF =4 INTF =3 FF =5 TF =0! TF =0! TF =1 FF = 0 FF = 0 TF = 3 INTF = 0 TF =1 FF = 1 INTF =3 FF = 0 TF = 3 FF = 0 TF = 4 FF =4 INTF = 0 TF = 4 INTF = 0 TF = 3 FF = 3 FF =5 INTF = 0 TF = 5 TF =0! CEE536 Example Problems 31 P.G. Ioannou & C. Srisuwanrat

32 3. ACTIVITY-ON-NODE SCHEDULING

33 Problem 3.1 A. Construct a precedence diagram. B. On the diagram, compute the four schedule dates (ESD, EFD, LSD, LFD) and the four floats (TF, FF, INTF, and IDF) for each activity, and the lag for each link. C. Identify the critical path No ACT DUR PREDECESSORS 5 B 5 10 M 4 B 15 N 9 B 0 Q 15 B 5 A 1 M,N 30 F 4 N,Q 35 X 9 Q 40 C 9 Q 45 Y 9 A,F,X 50 S 6 F 55 J 5 X,F 60 T 10 C 65 V 5 Y,S 70 U 10 V,T,J CEE536 Example Problems 33 P.G. Ioannou & C. Srisuwanrat

34 Problem 3. A. Construct a precedence diagram. B. On the diagram, compute the four schedule dates (ESD, EFD, LSD, LFD) and the four floats (TF, FF, INTF, and IDF) for each activity, and the lag for each link. C. Identify the critical path No ACT DUR PREDECESSORS 5 A 1 10 B 8 A 15 C 4 A 0 P 7 A 5 L B 30 M 4 C 35 Q 4 P,C 40 N 9 P 45 Y 5 L,Q 50 F 10 M 55 J Q 60 S N 65 V 5 Y,F,J 70 Q1 1 V,S CEE536 Example Problems 34 P.G. Ioannou & C. Srisuwanrat

35 Problem 3.3 A. Construct a precedence diagram. B. On the diagram, compute the four schedule dates (ESD, EFD, LSD, LFD) and the four floats (TF, FF, INTF, and IDF) for each activity, and the lag for each link. C. Identify the critical path No ACT DUR PREDECESSORS 5 B 5 10 M 4 B 15 N 9 B 0 X 15 B 5 A 5 M,N 30 F 6 N,X 35 Q X 40 C 4 X 45 Y 10 A 50 S 10 F,A 55 R Q,F 60 T 5 C,Q 65 K 7 Y,S,R 70 U 3 K,T CEE536 Example Problems 35 P.G. Ioannou & C. Srisuwanrat

36 Problem 3.4 A. Construct a precedence diagram. B. On the diagram, compute the four schedule dates (ESD, EFD, LSD, LFD) and the four floats (TF, FF, INTF, and IDF) for each activity, and the lag for each link. C. Identify the critical path No ACT DUR PREDECESSORS 5 A 9 10 B 1 A 15 C 10 A 0 P 10 A 5 L B,C 30 F 10 C,P 35 Q 8 P 40 N 6 P 45 Y 7 L 50 T 4 F,L 55 R 9 F,Q 60 S 1 N,Q 65 V 10 Y,T,R 70 U 1 V,S CEE536 Example Problems 36 P.G. Ioannou & C. Srisuwanrat

37 Solution M 4 5 A 1 45 Y ,5 14,5 8 14,14 9 0,0 9 0,0 38 0, B 5 15 N 9 30 F 4 50 S 6 65 V 5 0 0,0 5 0, ,0 5 11,0 5 5,0 9 5,0 3 8,8 38 0,3 38 0,0 43 0, Q X 9 55 J 5 70 U ,0 0 0,0 0 0,0 9 0,0 38 9,9 43 0,9 43 0,0 53 0, C 9 4 4,0 33 4,0 60 T ,4 43 0,0 CEE536 Example Problems 37 P.G. Ioannou & C. Srisuwanrat

38 Solution B 8 5 L 45 Y ,0 3,0 1 3,1 14,0 14, 19 0, A 1 15 C 4 30 M 4 50 F V 5 0 0,0 1 0,0 1 0,0 5 0,0 5 0,0 9 0,0 9 0,0 19 0,0 19 0,0 4 0, P 7 35 Q 4 55 J 70 Q1 1 3,0 10,0 10,0 14,0 17 5,5 19 0,3 4 0,0 5 0, N ,0 5,0 60 S 5,5 4 0,0 CEE536 Example Problems 38 P.G. Ioannou & C. Srisuwanrat

39 Solution M 4 5 A 5 45 Y ,5 7,5 1 7,0 6 7,0 6 7,0 36 7, B 5 15 N 9 30 F 6 50 S K 7 0 0,0 5 0,0 11 6,0 0 6,0 0 0,0 6 0,0 6 0,0 36 0,0 36 0,0 43 0, X Q 55 R 70 U 3 5 0,0 0 0,0 3 1, 34 10, 34 8,8 36 0,0 43 0,0 46 0, C , ,0 60 T , ,0 CEE536 Example Problems 39 P.G. Ioannou & C. Srisuwanrat

40 Solution B 1 5 L 45 Y 7 1 1,9 3,9 7 8,0 9 8,0 9 8,8 36 0, A 9 15 C F T 4 65 V ,0 9 0,0 9 0,0 19 0,0 3,0 3 3,0 3 3,3 36 0,0 36 0,0 46 0, P Q 8 55 R 9 70 U 1 9 0,0 19 0,0 19 0,0 7 0,0 7 0,0 36 0,0 46 0,0 47 0, N 6 1, 7 0, 60 S , ,18 CEE536 Example Problems 40 P.G. Ioannou & C. Srisuwanrat

41 Problem 3.5 CEE536 Example Problems 41 P.G. Ioannou & C. Srisuwanrat

42 Problem 3.6 CEE536 Example Problems 4 P.G. Ioannou & C. Srisuwanrat

43 Problem 3.7 CEE536 Example Problems 43 P.G. Ioannou & C. Srisuwanrat

44 Problem 3.8 CEE536 Example Problems 44 P.G. Ioannou & C. Srisuwanrat

45 Problem 3.9 (CalActivityOnNode) CEE536 Example Problems 45 P.G. Ioannou & C. Srisuwanrat

46 4. SCHEDULING USING A LINK MATRIX

47 Problem 4.1 Using matrix to calculate TF and FF NO ACT DUR PREDECESSORS 5 A 4 10 B 5 A 15 C 3 A 0 Q 8 A 5 M 1 B,C 30 L 8 C,B,Q 35 X Q,C 40 N 7 Q 45 Z 8 M,L 50 S 8 L,M 55 J 6 X,L 60 T 4 N 65 V 10 Z,S,J 70 Q1 6 V,T Problem 4. Using matrix to calculate TF and FF NO ACT DUR PREDECESSORS 5 A 7 10 B 5 A 15 C 7 A 0 P 5 A 5 M 1 B,C 30 L 5 C,B,P 35 N P,C 40 Y 9 P 45 G 5 M,L 50 S 9 L,M,N 55 R N 60 T 5 Y,N 65 K 9 G,S 70 Q1 5 K,T,R CEE536 Example Problems 47 P.G. Ioannou & C. Srisuwanrat

48 Problem 4.3 Using matrix to calculate TF and FF NO ACT DUR PREDECESSORS 5 A B 9 A 15 C 7 A 0 Q 9 A 5 L 1 B 30 F 5 C 35 X 4 Q,C 40 N 7 Q 45 Y 6 L 50 T 9 F,L 55 J 3 X,F,N 60 S 6 N,X 65 V 10 Y,T,J 70 U 8 V,S CEE536 Example Problems 48 P.G. Ioannou & C. Srisuwanrat

49 Solution 4.1 SUC A B C Q M L X N Z S J T V Q1 DUR ESD EFD FF A TF FF B TF FF 5 5 C TF FF Q TF FF M TF FF L TF FF 6 6 X TF 8 8 FF 0 0 N TF FF 0 0 Z TF 0 0 FF 0 0 S TF 0 0 FF J TF FF T TF FF 0 0 V TF 0 0 FF 0 Q1 44 TF 0 CEE536 Example Problems 49 P.G. Ioannou & C. Srisuwanrat

50 Solution 4. A B C P M L N Y G S R T K Q1 DUR ESD EFD FF A TF 0 0 FF B TF 6 FF C TF FF 0 0 P TF 5 11 FF M TF FF L TF FF N TF FF 0 0 Y TF FF 4 4 G TF 4 4 FF 0 0 S TF 0 0 FF R TF FF T TF FF 0 0 K TF 0 0 FF 0 Q1 4 TF 0 CEE536 Example Problems 50 P.G. Ioannou & C. Srisuwanrat

51 Solution 4.3 A B C Q L F X N Y T J S V U DUR ESD EFD FF A TF 0 0 FF 0 0 B TF FF 0 0 C TF FF Q TF 5 FF 0 0 L TF 5 FF F TF FF X TF FF N TF 9 FF 5 5 Y TF 5 5 FF 0 0 T TF 0 0 FF J TF FF 9 9 S TF 9 9 FF 0 0 V TF 0 0 FF 0 U 49 TF 0 CEE536 Example Problems 51 P.G. Ioannou & C. Srisuwanrat

52 5. PROJECT UPDATING

53 Project Updating Original Target Schedule B-8 E-6 G-4 A-3 C-5 F-3 H-6 K-3 D-3 Update Information at data date = Activity D takes 6 days to finish.. Activity F takes 5 more days to complete. 3. It is expected that it will take total 15 days to finish activity B. 4. Activity H cannot start until date 17 because of the delay of material B A-3 C-5 F1- Dummy 3 9 D-6 Please study these two networks and pay attention to on-processing activities (B and F) B-8 E-6 G Dummy F-5 Waiting Dummy H-6 K-3 CEE536 Example Problems 53 P.G. Ioannou & C. Srisuwanrat

54 6. PERT PNET

55 6. Probabilistic Scheduling using PERT and PNET Problem 6.1 Probabilistic scheduling using PERT and PNET methods 1.) Set up a table and calculate Early Event Times (TE), Late Event Times (TL), Activity Free Slack (AFS), and Activity Total Slack (ATS)..) Determine the PERT "critical path(s)" and the mean and standard deviation for the project duration. 3.) Use PNET method to find the project durations and their corresponding probabilities of project completion from 0 to 100%. i j Act a M b 5 10 A B C D E F G H I J K L M N O Z CEE536 Example Problems 55 P.G. Ioannou & C. Srisuwanrat

56 Problem 6. Probabilistic scheduling using PERT and PNET methods 1.) Set up a table and calculate Early Event Times (TE), Late Event Times (TL), Activity Free Slack (AFS), and Activity Total Slack (ATS)..) Determine the PERT "critical path(s)" and the mean and standard deviation for the project duration. 3.) Use PNET method to find the project durations and their corresponding probabilities of project completion from 0 to 100%. i j Act a M b 5 10 A B C Y U I O P L K J H G F D S CEE536 Example Problems 56 P.G. Ioannou & C. Srisuwanrat

57 Problem 6.3 Probabilistic scheduling using PERT and PNET methods 1.) Set up a table and calculate Early Event Times (TE), Late Event Times (TL), Activity Free Slack (AFS), and Activity Total Slack (ATS)..) Determine the PERT "critical path(s)" and the mean and standard deviation for the project duration. 3.) Use PNET method to find the project durations and their corresponding probabilities of project completion from 0 to 100%. i j Act a M b 5 10 S U P E R W O M A N X Y CEE536 Example Problems 57 P.G. Ioannou & C. Srisuwanrat

58 Solution 6.1 Step 1. Calculate E[ti], SD[ti], and Var[ti]. Table 1. Activity Properties i j Act a M b E[ti] SD[ti] Var[ti] TEi TEi+E[ti] TLi-E[ti] TLj ATS AFS 5 10 A B C D E F G H I J K L M N O Z E[ti] = (a+4m+b)/6 SD[ti] = (b-a)/6 => Var[ti] = SD[ti] CEE536 Example Problems 58 P.G. Ioannou & C. Srisuwanrat

59 Step. Determine all possible paths and calculate their E[T], Var[T], and SD[T] Table. Path Properties E[Ti] i path - E[Ti] Var[Ti] SD[Ti] 3SD[Ti] +3SD[Ti] Example: Path 1 consists of activities 5-15, 15-35, 35-50, and Activity => E[ti] E[T 1 ] = 4 Var[ti] Var[T 1 ] = 1.54 Thus, SD[T 1 ] of path1 is = 1.4 NOTE: 1. Do not add SD[ti] of activities to get SD[T] of a path. Path SD[T] must be derived from a square root of the summation of Var[ti] of activities in the path.. Path 6 and 7 can be neglected because their E[Ti]+3SD[Ti], which are and 17.81, are relatively short compared to the maximum E[T]-3SD[T], which is Paths in Table must be sorted by E[T] before constructing Table 3, otherwise you might represent a longer path by a shorter one. CEE536 Example Problems 59 P.G. Ioannou & C. Srisuwanrat

60 Step 3. Calculate correlation between paths Table 3. Correlations between paths r 1 Pij 0.41 r 1 3 Pij 0.48 r 1 4 Pij 0 r 1 5 Pij 0.45 r 1 6 Pij 0.18 r 1 7 Pij 0 r 3 Pij 0.38 r 4 Pij 0.4 r 5 Pij 0.41 r 6 Pij 0 r 7 Pij 0 r 3 4 Pij 0 r 3 5 Pij 0.75 r 3 6 Pij 0 r 3 7 Pij 0 r 4 6 Pij 0.3 r 4 7 Pij 0. r 6 7 Pij 0.6 Correlation greater than 0.5, thus eliminate path 5 ρ k ij σ Ti σ Tj = σ k ( πi π j ) σ σ k Ti Tj ( π π ) i j ρij is the correlation between path i and j. are activities that are in both path i and j. is standard deviation of path i, SD[Ti] is standard deviation of path j, SD[Tj] Example: Path 1 and have activities in common (its SD[ti] is 0.83), and their SD[T] are 1.4 and Thus, ρ = ij σ k ( πi π j ) σ σ k Ti Tj 0.83 = ρ = = 0.41 NOTE: 1. Paths in Table must be sorted by E[T] before constructing Table 3, otherwise you might represent a longer path by shorter one.. Path 5 is represented by path 3. ( A shorter path with high correlation to a longer path is represented by the longer one, NOT the other way around) CEE536 Example Problems 60 P.G. Ioannou & C. Srisuwanrat

61 Step 4. Calculate probabilities of project completion Table 4. Probability of Project Completion T 1 all combine (PERT) (PNET) x E[ Ti] F = i ( x) Fu σ i Example: 6 4 Probability of finishing the project less than 6 days according to path 1 is F 1 (6) = Fu = Fu(1.61) = Probability of finishing the project less than 19 days according to path 1 is 19 4 F 1(19) = Fu = Fu( 4) = 1 Fu(4) = Probability of finishing the project greater that 6 days is 1-F 1 (6) = = 0.54 NOTE: As shown in Table 4, path 6 and 7 can be ignored since their E[Ti]+3SD[Ti] (from Table ), which are and respectively,are less than the maximum E[Ti]-3SD[Ti], 0.8. In Table 4, agreeing to Table, probabilities of path 6 and 7 are all equal to 1. CEE536 Example Problems 61 P.G. Ioannou & C. Srisuwanrat

62 Probability of Project Completion Probability Day PERT PNET CEE536 Example Problems 6 P.G. Ioannou & C. Srisuwanrat

63 Solution 6. Step 1. Calculate E[ti], SD[ti], and Var[ti]. Table 1. Activity Properties i j Act a M b E[ti] SD[ti] Var[ti] TEi TEi+E[ti] TLi-E[ti] TLj ATS AFS 5 10 A B C Y U I O P L K J H G F D S Step. Determine all possible paths and calculate their E[T], Var[T], and SD[T] Table. Path Properties E[Ti] path i E[Ti] Var[Ti] SD[Ti] -3SD[Ti] +3SD[Ti] CEE536 Example Problems 63 P.G. Ioannou & C. Srisuwanrat

64 Step 3. Calculate correlation between paths Step 4. Calculate probabilities of project completion Table 3. Correlations between paths Table 4. Probability of Project Completion r 1 Pij 0.69 r 1 3 Pij 0.5 r 1 4 Pij 0.03 r 1 5 Pij 0.03 r 3 4 Pij 0.05 r 3 5 Pij 0.05 r 4 5 Pij 0.68 T 1 (PERT) 3 4 all combine (PNET) CEE536 Example Problems 64 P.G. Ioannou & C. Srisuwanrat

65 Problem 6.3 Table 1. i j Act a M b E[ti] SD[ti] Var[ti] TEi TEi+E[ti] TLi-E[ti] TLj ATS AFS 5 10 S U P E R W O M A N X Y Table. E[Ti] i PATH E[Ti] Var[Ti] SD[Ti] -3SD[Ti] +3SD[Ti] * * NOTE: 1. Data in Table must be sorted by E[T] before constructing Table 3, otherwise you might try to represent a longer path by shorter one.. Path 4 and 5 can be ignored since their E[Ti]+3SD[Ti], which are and respectively,are less than the maximum E[Ti]-3SD[Ti], 0.9. CEE536 Example Problems 65 P.G. Ioannou & C. Srisuwanrat

66 Table 3. Table 4. r 1 Pij 0.00 r 1 3 Pij 0.00 r 1 4 Pij 0.59 r 1 5 Pij 0.13 r 3 Pij 0.6 r 5 Pij 0.70 NOTE: Path 4 can be represented by path 1, and also path 3 can be represented by path. ( A shorter path with high correlation to a longer path is represented by the longer one, NOT the other way around) T 1 (PERT) all combine (PNET) CEE536 Example Problems 66 P.G. Ioannou & C. Srisuwanrat

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