Download - Advanced Models for Project Management L. Valadares Tavares J. Silva Coelho IST, Lisbon, 2002
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Advanced Models for Project Management
L. Valadares TavaresJ. Silva Coelho
IST, Lisbon, 2002
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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
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The cycle of development of an organization
Mission
Objectives
Goals
Externalenvironmen
t
Internal conditions
Strategies
Plans and programs
PROJECTS
Appraisal, monitoring,
Control
Results and Evaluations
Needs
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An hierarchical decomposition of the project into activities
Project
Level 1
Level 2
1.1 1.2 1.3
1. (N-1)
1.N1
2.1.1 2.1.2 2.2.1 2.2.2 2.2.3 2.3.1 2.N.1
2.N.2
. . .
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Project Definition a) activities: b) precedences:
Where:
c) attributes: q=1: duration (D) q=2: cost (C) q=3: resource 1, ... (R1,...)
NiA ,...,1
NiJ ,...,1
jiNji JiJj '',...,1,
lliNlji JjJiJj ,...,1,,
NiQqB ,...,1,,...,1
lJliiNi JjJjjJi
':' ',...,1
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Directed Acyclic Graph
AiJi Li
jiNi JijL :,...,1
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AoN vs AoA
1 12 13
Start:Node S
2 5
4
3
10
11
9
i = 6End:
Node E
7 8
x
S
1
2
47
12 13
8 11
95
3
6
10
E
dummy activity
AoN
AoA
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Different Precedences, i->j 1) F -> S
2) S -> F
3) F -> F
4) S -> S
iij DSS
jij DSS
jiij DDSS
ij SS
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Different Unions
a
d
c
b
a
b
c
d
a
b
d
c
Intersection Inclusive union Exclusive union
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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 ww CkCkS
)...,...( 11...1 NNwi AAAALC
kCTTAALT SCkiiNi )(#,, ),(...1
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Example Instance:
Possible assignments T:
757565654321 ,,1,,,1,,,1,,,1,,,,,4 AAAAAAAAAAAAS
7654321 ,,,,,, AAAAAAAT 7654321 ,,,,,, AAAAAAAT
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Resources
S (t)
CumulativeConsumption
Time
Start of theProject
End of theProject
R (t)
Time
Capacity curveC (t)
A0
A1
Non-renewable Renewable
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Earliest and latest starting times of the activities
12
1
13
4 7 8
11
E
103
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
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C(i) in terms of D(i)
CostC (i)
Duration D(i)Min C(i)=mi Max C(i)=Mi
Reduction of D(i)CPMi
D
C
minimal
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Structural Modeling Project Hardness
Project Complexity A: arcs N: nodes
A/N 2(A-N+1)/(N-1)(N-2) A2/N
N
iiJN
H1
'#1 1;0
1
2
N
Hh
N
iiJN
CI1
#1
Pascoe, 1966
Davies, 1974
Kaimann, 1974
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Hierarchical Levels a) Progressive level
b) Regressive level
1)(max
1)( jpJ
Jip
iJji
i
1)(min
)(max)(
...1
jqL
jpLiq
iLji
Nji
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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
12 1
31
4
2 5 6
9
3
10
7 8
11
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Adjacency Matrix Aij
1 if there is a direct precedence i->j
0 if not
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Level Adjacency Matrix Xij – number of
precendences links between level i and j
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Example
2
3 4 5 10
9
876
1
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Morphology and Simulation of Project Networks a) Series-network
b) Parallel-network
. . .
0
i=1 i=N
N+1
i=1
i=N
.
.
.
N
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Morphologic Indicators 1
NI 1 )(max...1
ipMNi
1;011
1
11
2
N
N
MN
I
aipNiaWMa )(:...1#)(...1
1;0
1)(max
1)(min1)(max
11)(max
...1
...1
...1
...1
3
aW
aWaW
aW
I
Ma
Ma
Ma
Ma
Size of problem
Serial/parallel
Activity distribution
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Morphologic Indicators 2
)()(max,...1
jpipViJjNi
vjpipJjNijivn iVv )()(:...1),,(#)(...1
Ma
aWaWD...2
)()1(
1;0
)1(
)1()1()1(
1)1(
4
WND
WNnWND
WND
I
Short direct precedences
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Morphologic Indicators 3
1;0
2
12
01
6
M
VM
MI
1;021
...1
)()(7
Ni
iqip
NI
Vv
vnTDP...1
)(
1;0)(2
...1
)1(
5
TDP
vnI Vv
v
Long direct precedences
Maximal direct precedences
Morphological float
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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
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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
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Distribution of tk in terms of I1 for the normal case
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Distribution of tk in terms of I1 for the exponential case
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Distribution of tk in terms of I2 for the normal case
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Distribution of tk in terms of I2 for the Exponential case
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Distribution of tk in terms of I4 for the normal case
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Distribution of tk in terms of I4 for the exponential case
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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...
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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
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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
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Lower Bound Maximal value of all lower bounds
(super optima) Ignoring resources (CPM) Ignoring activities (for each resource):
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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
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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
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Latest Starting Time, and AoN