Advanced Models for Project Management

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

4. 2.2.1 2.2.2 2.2.3 Project . . . Level 1 1.N1 1.1 1.2 1.3 1. (N-1) Level 2 2.1.1 2.1.2 2.3.1 2.N.1 2.N.2 An hierarchical decomposition of the project into activities

5. Project Definition • a) activities: • b) precedences: • Where: • c) attributes: • q=1: duration (D) • q=2: cost (C) • q=3: resource 1, ... (R1,...)

6. Directed Acyclic Graph Ai Ji Li

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

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

9. a a b b d d a d b c c c Different Unions Intersection Inclusive union Exclusive 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. Capacity curve C (t) Cumulative Consumption R (t) S (t) End of the Project Start of the Project Time A0 A1 Time Resources Non-renewable Renewable

13. 15 16 10 11 0 21 9 30 37 37 31 31 7 24 27 27 21 21 21 25 13 13 10 10 0 0 0 14 13 12 1 7 4 8 E 6 11 5 2 S 9 3 10 Earliest and latest starting times of the activities 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. Cost C (i) Duration D(i) Min C(i)=mi Max C(i)=Mi C(i) in terms of D(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) • A2/N Pascoe, 1966 Davies, 1974 Kaimann, 1974

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

17. 2 5 5 4 4 4 2 3 3 0 0 1 5 5 2 6 6 1 1 3 4 4 7 7 7 1 3 2 2 3 3 4 4 4 5 5 6 6 6 12 13 7 8 1 11 4 5 2 6 9 3 10 Progressive and Regressive levels

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

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

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