1. Advanced Models for Project Management L. Valadares Tavares J. Silva Coelho IST, Lisbon, 2002

2. Contents 1. A systemic introduction to project management 2. Basic models for project management 3. Structural modelling of project networks 4. Morphology and simulation of project networks 5. Duration of projects 6. Scheduling of project networks 7. The assessment and evaluation of projects 8. The optimal scheduling of a project in terms of its duration

3. The cycle of development of an organization Mission Objectives Goals External environmen t Internal conditions Strategies Plans and programs PROJECTS Appraisal, monitoring, Control Results and Evaluations Needs

4. An hierarchical decomposition of the project into activities Project Level 1 Level 2 1.11.21.3 1. (N-1) 1.N1 2.1.12.1.22.2.12.2.22.2.32.3.12.N.1 2.N.2...

5. Project Definition a) activities: b) precedences: Where: c) attributes: q=1: duration (D) q=2: cost (C) q=3: resource 1,... (R 1,...)

6. Directed Acyclic Graph AiAi JiJi LiLi

7. AoN vs AoA 1 12 13 Start: Node S 2 5 4 3 10 11 9 i = 6 End: Node E 7 8 x S 1 2 4 7 12 13 8 11 9 5 3 6 10 E dummy activity AoN AoA

8. Different Precedences, i->j 1) F -> S 2) S -> F 3) F -> F 4) S -> S

9. Different Unions a d c b a b c d a b d c IntersectionInclusive unionExclusive union

10. Statisfiability problem Conjuntion of disjunctions of variables Activities are boolean variables, if true the activity is realized, if false is not SATK: k is an integer Find an assignment T:

11. Example Instance: Possible assignments T:

12. Resources S (t) Cumulative Consumption Time Start of the Project End of the Project R (t) Time Capacity curve C (t) A0A0 A1A1 Non-renewableRenewable

13. Earliest and latest starting times of the activities 12 1 13 4 7 8 11 E 10 3 S 2 5 6 9 0 21 9 30 37 37 31 31 7 24 0 14 21 21 21 25 13 13 10 10 0 0 10 11 15 16 27 27 Activity Duration 1 10 2 3 3 7 4 5 5 8 6 2 7 11 8 4 9 6 10 7 11 6 12 9 13 7

14. C(i) in terms of D(i) Cost C (i) Duration D(i) Min C(i)=m i Max C(i)=M i Reduction of D(i) minimal

15. Structural Modeling Project Hardness Project Complexity A: arcs N: nodes A/N 2(A-N+1)/(N-1)(N-2) A 2 /N Pascoe, 1966 Davies, 1974 Kaimann, 1974

16. Hierarchical Levels a) Progressive level b) Regressive level

17. Progressive and Regressive levels 4 4 4 2 3 3 0 0 1 5 5 2 6 6 1 1 3 4 4 7 7 7 6 6 6 2 5 5 1 3 2 2 3 3 5 5 4 4 4 1212 1313 1 4 2 5 6 9 3 1010 7 8 1

18. Adjacency Matrix Aij 1 if there is a direct precedence i->j 0 if not

19. Level Adjacency Matrix Xij – number of precendences links between level i and j

20. Example 2 3 4 5 10 9 8 7 6 1

21. Morphology and Simulation of Project Networks a) Series-network b) Parallel-network... 0 i=1 i=N N+1 i=1 i=N...... N

22. Morphologic Indicators 1 Size of problem Serial/parallel Activity distribution

23. Morphologic Indicators 2 Short direct precedences

24. Morphologic Indicators 3 Long direct precedences Maximal direct precedences Morphological float

25. Example N=10, M=5, V=4, D=16, n(1)=8, TDP=16 I1=10, I2=0.44, I3=1, I4=0, I5=0.66, I6=1, I7=0.74

26. Duration of Projects Uncertain duration of activities Each activity is assumed to follow a distribution Goal: find total project duration distribution Solution Simulating durations for activities and calculate the total project duration for each simulation tk = simulation total duration / deterministic total duration

27. Distribution of tk in terms of I1 for the normal case

28. Distribution of tk in terms of I1 for the exponential case

29. Distribution of tk in terms of I2 for the normal case

30. Distribution of tk in terms of I2 for the Exponential case

31. Distribution of tk in terms of I4 for the normal case

32. Distribution of tk in terms of I4 for the exponential case

33. Optimal Scheduling The Resource Constained Project Scheduling Problem (RSPSP): Instance: set of activities, and for each activity a set of precedences, a duration and resource usage. For each resource exist a resource capacity limit. Goal: Find a the optimal valid schedule, that is a start time for each activity that: Does not violate precedence constraints Does not violate resource limit capacity RCPSP contains several problems, like Jobshop, Flowshop, Openshop, Binpacking...

34. PSS/SSS Schedule Parallel Scheduling Scheme Process each instant t, starting at 0 Schedule for starting at t the most important activity that can start at t If no more activities can start at t, increment t PSS: no delay schedule, can eventually not contain any optimal schedule Serial Scheduling Scheme Select activities by order of importance, not violating precedence constraints Schedule the activity to the first instant that can start SSS: active schedule, contain at least one optimal schedule

35. Priority Rules Importance of activities Latest Start Time (LST) Latest Finish Time (LFT) Shortest Processing Time (SPT) Greatest Rank Positional Weight (GRPW) Sum processing time and also the time of direct successors Most Total Successors (MTS) Count all successors, direct or indirect Most Total Successors Processing Time (MTSPT) Sum all processing time of all sucessors, direct or indirect

36. Lower Bound Maximal value of all lower bounds (super optima) Ignoring resources (CPM) Ignoring activities (for each resource):

37. Looking for the best solution Meta-Heuristics Sampling Method Local Search Local search with restart Simulated annealing Tabu-search Genetic Algorithms Can deal with large instances Exact methods Branch-and-Bound Have the optimal solution after finish

38. Example Available resources per time unit: L=3, T=4 LST: 2; 1; 3; 4; 5; 6; 7; 8; 13; 10; 11; 12; 14; 9

39. Latest Starting Time, and AoN