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
The cycle of development of an organization
Mission
Needs
Strategies
Objectives
Plans and programs
PROJECTS
Goals
Appraisal, monitoring, Control
Results and Evaluations
4
An hierarchical decomposition of the project into
activities
5
Project Definition

• a) activities
• b) precedences
• Where
• c) attributes
• q1 duration (D)
• q2 cost (C)
• q3 resource 1, ... (R1,...)

6
Directed Acyclic Graph
Ai
Ji
Li
7
AoN vs AoA
AoA
AoN
8
Different Precedences, i-gtj

• 1) F -gt S
• 2) S -gt F
• 3) F -gt F
• 4) S -gt S

9
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
Resources
Non-renewable
Renewable
13
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
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-N1)/(N-1)(N-2)
• A2/N

Pascoe, 1966
Davies, 1974
Kaimann, 1974
16
Hierarchical Levels

• a) Progressive level
• b) Regressive level

17
Progressive and Regressive levels
18
Adjacency Matrix

• Aij
• 1 if there is a direct precedence i-gtj
• 0 if not

19
Level Adjacency Matrix

• Xij number of precendences links between level
  i and j

20
Example
21
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

• N10, M5, V4, D16, n(1)8, TDP16
• I110, I20.44, I31, I40, I50.66, I61, I70.74

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 L3, T4
LST 2 1 3 4 5 6 7 8 13 10 11 12 14 9
39
Latest Starting Time, and AoN