1. Chapter 10

2. Activity-on-Node Network Fundamentals X Y Z Y and Z are preceded by X Y and Z can begin at the same time, if you wish (B) ABC A is preceded by nothing B is preceded by A C is preceded by B (A) J K L M J, K, & L can all begin at the same time, if you wish (they need not occur simultaneously) All (J, K, L) must be completed before M can begin but XZ AA Y (C) (D) Z is preceded by X and Y AA is preceded by X and Y

3. Network-Planning Models A project is made up of a sequence of activities that form a network representing a project. The path taking longest time through this network of activities is called the “critical path.” The critical path provides a wide range of scheduling information useful in managing a project. Critical Path Method (CPM) helps to identify the critical path(s) in the project networks.

4. Prerequisites for Critical Path Methodology A project must have: well-defined jobs or tasks whose completion marks the end of the project; independent jobs or tasks; and tasks that follow a given sequence.

5. Types of Critical Path Methods CPM with a Single Time Estimate –Used when activity times are known with certainty. –Used to determine timing estimates for the project, each activity in the project, and slack time for activities. CPM with Three Activity Time Estimates –Used when activity times are uncertain. –Used to obtain the same information as the Single Time Estimate model and probability information. Time-Cost Models –Used when cost trade-off information is a major consideration in planning. –Used to determine the least cost in reducing total project time.

6. Steps in the CPM with Single Time Estimate 1. Activity Identification. 2. Activity Sequencing and Network Construction. 3. Determine the critical path. –From the critical path all of the project and activity timing information can be obtained.

7. Example 1. CPM with Single Time Estimate Consider the following consulting project: Develop a critical path diagram and determine the duration of the critical path and slack times for all activities ActivityDesignationImmed. Pred.Time (Weeks) Assess customer's needsANone2 Write and submit proposalBA1 Obtain approvalCB1 Develop service vision and goalsDC2 Train employeesEC5 Quality improvement pilot groupsFD, E5 Write assessment reportGF1

8. Example 1: First draw the network A(2)B(1) C(1) D(2) E(5) F(5) G(1) ANone2 BA1BA1 CB1CB1 DC2DC2 EC5EC5 FD,E5 GF1GF1 Act.Imed. Pred. Time

9. Example 1: Determine early starts and early finish times ES = ? EF = ? ES=0 EF=2 ES=2 EF=3 ES=3 EF=4 ES=4 EF=9 ES=4 EF=6 A(2)B(1) C(1) D(2) E(5) F(5) G(1)

10. Example 1: Determine early starts and early finish times ES=9 EF=14 ES=14 EF=15 ES=0 EF=2 ES=2 EF=3 ES=3 EF=4 ES=4 EF=9 ES=4 EF=6 A(2)B(1) C(1) D(2) E(5) F(5) G(1) WHAT IS EF OF THE PROJECT?

11. Example 1: Determine late starts and late finish times ES=14 EF=15 A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14 LF=15 Duration = 15 weeks

12. Example 1: Determine late starts and late finish times ES=14 EF=15 A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14 LF=15 Duration = 15 weeks

13. Example 1: Determine late starts and late finish times A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14 LF=15 LS=9 LF=14 LS=4 LF=9 LS=7 LF=9 LS = ? LF = ?

14. Example 1: Determine late starts and late finish times A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14 LF=15 LS=9 LF=14 LS=4 LF=9 LS=7 LF=9 LS=3 LF=4 LS=2 LF=3 LS=0 LF=2

15. Example 1: DON’T WRITE DOWN, JUST TO SHOW ALL NUMBERS ES=9 EF=14 ES=14 EF=15 ES=0 EF=2 ES=2 EF=3 ES=3 EF=4 ES=4 EF=9 ES=4 EF=6 A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14 LF=15 LS=9 LF=14 LS=4 LF=9 LS=7 LF=9 LS=3 LF=4 LS=2 LF=3 LS=0 LF=2

16. NOW::: FOR CRITICAL PATH

17. Example 1: Critical Path & Slack A(2)B(1) C(1) D(2) E(5) F(5) G(1) ALL THAT IS NEEDED ES & LS or EF & LF I PREFER ES & LS

18. Example 1: Critical Path & Slack ES=9ES=14ES=0ES=2ES=3 ES=4 A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14LS=9 LS=4 LS=7 LS=3LS=2LS=0 Duration = 15 weeks

19. Example 1: Critical Path & Slack ES=9ES=14ES=0ES=2ES=3 ES=4 A(2)B(1) C(1) D(2) E(5) F(5) G(1) LS=14LS=9 LS=4 LS=7 LS=3LS=2LS=0 Slack=(7-4 = 3 Wks A CHECK TASK LS - ES CP A 0 - 0 YES B 2 - 2 YES C 3 - 3 YES D 7 - 4 NO E 4 - 4 YES F 9 - 9 YES G 14 - 14 YES THEREFORE CP = A-B-C-E-F-G

20. Example 2. CPM with Three Activity Time Estimates a m b

21. Example 2. Expected Time Calculations ET(A)= 3+4(6)+15 6 ET(A)=42/6=7

22. Example 2. Network A(7) B (5.333) C(14) D(5) E(11) F(7) H(4) G(11) I(18) Duration = 54 Days

23. Example 2. Network A(7) B (5.333) C(14) D(5) E(11) F(7) H(4) G(11) I(18) Duration = 54 Days ES=0 ES=7 ES=21 ES=32 ES=12 ES=36 ES=0ES=5.333 LS=36 LS=32 LS=21LS=7 LS=0 LS=25 LS=20 LS=25LS=19.667

24. THEREFORE: CRITICAL PATH IS: A-C-E-H-I

25. Example 2. Probability Exercise What is the probability of finishing this project in less than 53 days? p(t < D) T E = 54 t D=53

26. (Sum the variance along the critical path.)

27. There is a 43.6% probability that this project will be completed in less than 53 weeks. p(Z < -.16) =.5 -.0636 =.436, or 43.6 % (Appendix D) T E = 54 p(t < D) t D=53 or -.16 Std Normal Dist.

28. Example 2. Additional Probability Exercise What is the probability that the project duration will exceed 56 weeks?

29. Example 2. Additional Exercise Solution t T E = 54 p(t > D) D=56 p(Z >.31) =.5 -.1217 =.378, or 37.8 % (Appendix D) or.31 Std Normal Dist.

30. Time-Cost Models Basic Assumption: Relationship between activity completion time and project cost. Time Cost Models: Determine the optimum point in time-cost tradeoffs. – Activity direct costs. – Project indirect costs. – Activity completion times.

31. CPM Assumptions/Limitations Project activities can be identified as entities. (There is a clear beginning and ending point for each activity.) Project activity sequence relationships can be specified and networked. Project control should focus on the critical path. The activity times follow the beta distribution, with the variance of the project assumed to equal the sum of the variances along the critical path. Project control should focus on the critical path.