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Step by step network construction and (cpm) calculations

The document describes the step-by-step process of constructing a network diagram and performing critical path method (CPM) calculations for a project. An example project is provided with 10 activities and their durations. The steps shown include: 1) Drawing the activity-on-arrow (AOA) network diagram for the example project. 2) Calculating the critical path through the network using forward and backward passes to determine earliest and latest start/finish times. 3) The forward pass calculates earliest start times while the backward pass calculates latest completion times. The example helps illustrate how to set up and analyze a CPM network diagram and critical path.

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Step by Step Network Construction and (CPM) Calculations Project Management Dr. Mahmoud Abbas Mahmoud Asst. Prof. 2016
Step by Step Network Construction and (CPM) Calculations Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Example* Draw the (AOA) network diagram for the activities given in the table, find the critical path (CP), and give all (CPM) calculations (complete network analysis table). *This Example is adapted from “Operation Research - An Introduction - 8th Edition - H.A. Taha (2007) These steps will be illustrated with an Example. Dr. Mahmoud Abbas Mahmoud
Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Note: Practically, nodes numbering can be start by (0) or (1). Dr. Mahmoud Abbas Mahmoud
A 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
A 0 1 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 1 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C D Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C D E Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E G F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E 4 G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud ???????
B A 0 2 1 C 3 D E 4 G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud Use Dummy !
B A 0 2 1 C 3 D E 4 G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E 4 I G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E 4 5 I G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E 4 5 I J G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B A 0 2 1 C 3 D E 4 5 6 I J G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 (AOA) Network Drawing is Completed Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 Critical Path Calculations Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 Forward Pass: Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 Forward Pass: Calculating the Earliest Start Time (ES): Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 Forward Pass: Calculating the Earliest Start Time (ES): Earliest Start Time (ES)j=Max [ (ES)i + Dij ] Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud NOTE: (ES) for Start Event = (0) Is NOT RELATED with the Number Given to Start Event (in Circle). Earliest Start Time (ES) for Start Event = (0)
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 2 0 [2+0]= 2 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud [0+3]= 3
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 Max [(3+3) , (2+2)] = max [6,4]= 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 max[(3+2) , (6+0)] = max [5,6] = 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Max [(6+3) ,(6+7)] =max [9,13] =13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 13 Max [(6+2),(13+6),(6+5]= Max [8,19,9] = 19 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Backward Pass: Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ 19 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Backward Pass: Calculating the Latest Completion Time (LC): Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ 19 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Backward Pass: Calculating the Latest Completion Time (LC): Latest Completion Time (LC)i=Min [ (LC)j - Dij ] Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ 19 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 19 13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 19 13 13 19 – 6 =13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 6 19 19 13 13 Min [(19-5),(13-7)= Min[14,6] = 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 6 6 19 19 13 13 Min [(19-2),(13-3),(6-0)] = Min [17,10,6] = 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 3 2 0 6 6 6 6 19 19 13 13 Min [(6-2),(6-3)] = Min [4, 3] = 3 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 3 3 4 2 0 6 6 6 6 19 19 13 13 6-2 = 4 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Min [(3-3),(4-2)] = Min [0, 2] = 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Finding the Critical Activities and the Critical Path Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 To Identify the Critical Activities Remember the Three Conditions: For a critical activity i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 The Critical Activities are: (0, 2), (2, 3), (3, 4), (4, 5) and (5, 6) The Critical Path is: (0 – 2 – 3 – 4 – 5 – 6) Project Completion Time is: 19 weeks Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Total Float (TF) and Free Float (FF) Dr. Mahmoud Abbas Mahmoud TFij = (LC)j – (ES)i – Dij FFij = (ES)j – (ES)i – Dij
Notice that only a critical activity can have a Zero Total Float (TF) B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 For activity H (4, 5) TF = 13 – 6 – 7 = 0 * FF = 13 – 6 – 7 = 0 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 For activity H (4, 5) TF = 13 – 6 – 7 = 0 * FF = 13 – 6 – 7 = 0 For activity I (4, 6) TF = 19 – 6 – 5 = 8 FF = 19 – 6 – 5 = 8 Dr. Mahmoud Abbas Mahmoud
B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 For activity H (4, 5) TF = 13 – 6 – 7 = 0 * FF = 13 – 6 – 7 = 0 For activity I (4, 6) TF = 19 – 6 – 5 = 8 FF = 19 – 6 – 5 = 8 For activity J (5, 6) TF = 19 – 13 – 6 = 0 * FF = 19 – 13 – 6 = 0 Dr. Mahmoud Abbas Mahmoud
Dr. Mahmoud Abbas Mahmoud We can notice from these results that all critical activities must have zero free floats but not each activity that has zero free float must be critical. For example, activity A (0, 1) has a zero Free Float but it is not critical.
Earliest Completion (EC) and Latest Start (LS) Times Activity Symbol Activity (i, j) Duration Dij (ES)i ECij = (ES)i + Dij LSij = (LC)j - Dij (LC)j A (0, 1) 2 0 2 2 4 B (0, 2) 3 0 3 0 3 C (1, 3) 2 2 4 4 6 D (2, 3) 3 3 6 3 6 E (2, 4) 2 3 5 4 6 Dummy (3, 4) 0 6 6 6 6 F (3, 5) 3 6 9 10 13 G (3, 6) 2 6 8 17 19 H (4, 5) 7 6 13 6 13 I (4, 6) 5 6 11 14 19 J (5, 6) 6 13 19 13 19 Dr. Mahmoud Abbas Mahmoud
Summary of (CPM) Calculations Activity Symbol Activity (i, j) Duration Dij (Es)i ECij LSij (LC)j TFij FFij A (0, 1) 2 0 2 2 4 2 0 B (0, 2) 3 0 3 0 3 0* 0 C (1, 3) 2 2 4 4 6 2 2 D (2, 3) 3 3 6 3 6 0* 0 E (2, 4) 2 3 5 4 6 1 1 Dummy (3, 4) 0 6 6 6 6 0* 0 F (3, 5) 3 6 9 10 13 4 4 G (3, 6) 2 6 8 17 19 11 11 H (4, 5) 7 6 13 6 13 0* 0 I (4, 6) 5 6 11 14 19 8 8 J (5, 6) 6 13 19 13 19 0* 0 Notice that only a critical activity can have a Zero total float (TF) values are marked with (*) mark to indicate the critical activities. Dr. Mahmoud Abbas Mahmoud
Dr. Mahmoud Al-Naimi

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Step by step network construction and (cpm) calculations

  • 1.
    Step by Step NetworkConstruction and (CPM) Calculations Project Management Dr. Mahmoud Abbas Mahmoud Asst. Prof. 2016
  • 2.
    Step by Step NetworkConstruction and (CPM) Calculations Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Example* Draw the (AOA) network diagram for the activities given in the table, find the critical path (CP), and give all (CPM) calculations (complete network analysis table). *This Example is adapted from “Operation Research - An Introduction - 8th Edition - H.A. Taha (2007) These steps will be illustrated with an Example. Dr. Mahmoud Abbas Mahmoud
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    B A 0 2 1 C 3 D E 4 56 I J G H F Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 21.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 (AOA) Network Drawing is Completed Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 22.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 Critical Path Calculations Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 23.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 Forward Pass: Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 24.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 Forward Pass: Calculating the Earliest Start Time (ES): Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 25.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 Forward Pass: Calculating the Earliest Start Time (ES): Earliest Start Time (ES)j=Max [ (ES)i + Dij ] Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 26.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud NOTE: (ES) for Start Event = (0) Is NOT RELATED with the Number Given to Start Event (in Circle). Earliest Start Time (ES) for Start Event = (0)
  • 27.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 28.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 2 0 [2+0]= 2 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 29.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud [0+3]= 3
  • 30.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 Max [(3+3) , (2+2)] = max [6,4]= 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 31.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 max[(3+2) , (6+0)] = max [5,6] = 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 32.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Max [(6+3) ,(6+7)] =max [9,13] =13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 33.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 13 Max [(6+2),(13+6),(6+5]= Max [8,19,9] = 19 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 34.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Backward Pass: Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ 19 Dr. Mahmoud Abbas Mahmoud
  • 35.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Backward Pass: Calculating the Latest Completion Time (LC): Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ 19 Dr. Mahmoud Abbas Mahmoud
  • 36.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 13 Backward Pass: Calculating the Latest Completion Time (LC): Latest Completion Time (LC)i=Min [ (LC)j - Dij ] Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ 19 Dr. Mahmoud Abbas Mahmoud
  • 37.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 38.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 19 13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 39.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 19 19 13 13 19 – 6 =13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 40.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 6 19 19 13 13 Min [(19-5),(13-7)= Min[14,6] = 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 41.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 2 0 6 6 6 6 19 19 13 13 Min [(19-2),(13-3),(6-0)] = Min [17,10,6] = 6 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 42.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 3 2 0 6 6 6 6 19 19 13 13 Min [(6-2),(6-3)] = Min [4, 3] = 3 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 43.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 3 3 4 2 0 6 6 6 6 19 19 13 13 6-2 = 4 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 44.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Min [(3-3),(4-2)] = Min [0, 2] = 0 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 45.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 46.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Finding the Critical Activities and the Critical Path Dr. Mahmoud Abbas Mahmoud
  • 47.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 To Identify the Critical Activities Remember the Three Conditions: For a critical activity i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Dr. Mahmoud Abbas Mahmoud
  • 48.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 49.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 50.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 51.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 52.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 53.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Critical Activity Conditions: i) (LC)i = (ES)i ii) (LC)j = (ES)j iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij Duration (week) Preceding Activity Activity 2-A 3-B 2AC 3BD 2BE 3C,DF 2C,DG 7C,D,EH 5C,D,EI 6H,FJ Dr. Mahmoud Abbas Mahmoud
  • 54.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 The Critical Activities are: (0, 2), (2, 3), (3, 4), (4, 5) and (5, 6) The Critical Path is: (0 – 2 – 3 – 4 – 5 – 6) Project Completion Time is: 19 weeks Dr. Mahmoud Abbas Mahmoud
  • 55.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Total Float (TF) and Free Float (FF) Dr. Mahmoud Abbas Mahmoud TFij = (LC)j – (ES)i – Dij FFij = (ES)j – (ES)i – Dij
  • 56.
    Notice that onlya critical activity can have a Zero Total Float (TF) B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 5 6 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 Dr. Mahmoud Abbas Mahmoud
  • 57.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 Dr. Mahmoud Abbas Mahmoud
  • 58.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 Dr. Mahmoud Abbas Mahmoud
  • 59.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 Dr. Mahmoud Abbas Mahmoud
  • 60.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 Dr. Mahmoud Abbas Mahmoud
  • 61.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 Dr. Mahmoud Abbas Mahmoud
  • 62.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 Dr. Mahmoud Abbas Mahmoud
  • 63.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 Dr. Mahmoud Abbas Mahmoud
  • 64.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 Dr. Mahmoud Abbas Mahmoud
  • 65.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 For activity H (4, 5) TF = 13 – 6 – 7 = 0 * FF = 13 – 6 – 7 = 0 Dr. Mahmoud Abbas Mahmoud
  • 66.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 For activity H (4, 5) TF = 13 – 6 – 7 = 0 * FF = 13 – 6 – 7 = 0 For activity I (4, 6) TF = 19 – 6 – 5 = 8 FF = 19 – 6 – 5 = 8 Dr. Mahmoud Abbas Mahmoud
  • 67.
    B=3 A=2 0 2 1 C=2 3 D=3 E=2 4 56 I=5 J=6 G=2 H=7 F=3 0 0 3 3 4 2 0 6 6 6 6 19 19 13 13 For activity A (0, 1) TF = 4 – 0 – 2 = 2 FF = 2 – 0 – 2 = 0 For activity B (0, 2) TF = 3 – 0 – 3 = 0 * FF = 3 – 0 – 3 = 0 For activity C (1, 3) TF = 6 – 2 – 2 = 2 FF = 6 – 2 – 2 = 2 For activity D (2, 3) TF = 6 – 3 – 3 = 0 * FF = 6 – 3 – 3 = 0 For activity E (2, 4) TF = 6 – 3 – 2 = 1 FF = 6 – 3 – 2 = 1 For Dummy activity (3, 4) TF = 6 – 6 – 0 = 0 * FF = 6 – 6 – 0 = 0 For activity F (3, 5) TF = 13 – 6 – 3 = 4 FF = 13 – 6 – 3 = 4 For activity G (3, 6) TF = 19 – 6 – 2 = 11 FF = 19 – 6 – 2 = 11 For activity H (4, 5) TF = 13 – 6 – 7 = 0 * FF = 13 – 6 – 7 = 0 For activity I (4, 6) TF = 19 – 6 – 5 = 8 FF = 19 – 6 – 5 = 8 For activity J (5, 6) TF = 19 – 13 – 6 = 0 * FF = 19 – 13 – 6 = 0 Dr. Mahmoud Abbas Mahmoud
  • 68.
    Dr. Mahmoud AbbasMahmoud We can notice from these results that all critical activities must have zero free floats but not each activity that has zero free float must be critical. For example, activity A (0, 1) has a zero Free Float but it is not critical.
  • 69.
    Earliest Completion (EC)and Latest Start (LS) Times Activity Symbol Activity (i, j) Duration Dij (ES)i ECij = (ES)i + Dij LSij = (LC)j - Dij (LC)j A (0, 1) 2 0 2 2 4 B (0, 2) 3 0 3 0 3 C (1, 3) 2 2 4 4 6 D (2, 3) 3 3 6 3 6 E (2, 4) 2 3 5 4 6 Dummy (3, 4) 0 6 6 6 6 F (3, 5) 3 6 9 10 13 G (3, 6) 2 6 8 17 19 H (4, 5) 7 6 13 6 13 I (4, 6) 5 6 11 14 19 J (5, 6) 6 13 19 13 19 Dr. Mahmoud Abbas Mahmoud
  • 70.
    Summary of (CPM)Calculations Activity Symbol Activity (i, j) Duration Dij (Es)i ECij LSij (LC)j TFij FFij A (0, 1) 2 0 2 2 4 2 0 B (0, 2) 3 0 3 0 3 0* 0 C (1, 3) 2 2 4 4 6 2 2 D (2, 3) 3 3 6 3 6 0* 0 E (2, 4) 2 3 5 4 6 1 1 Dummy (3, 4) 0 6 6 6 6 0* 0 F (3, 5) 3 6 9 10 13 4 4 G (3, 6) 2 6 8 17 19 11 11 H (4, 5) 7 6 13 6 13 0* 0 I (4, 6) 5 6 11 14 19 8 8 J (5, 6) 6 13 19 13 19 0* 0 Notice that only a critical activity can have a Zero total float (TF) values are marked with (*) mark to indicate the critical activities. Dr. Mahmoud Abbas Mahmoud
  • 71.