Advanced Models for Project Management

L. Valadares TavaresJ. Silva Coelho

IST, Lisbon, 2002

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

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

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

. . .

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

Directed Acyclic Graph

AiJi Li

jiNi JijL :,...,1

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

Different Precedences, i->j 1) F -> S

2) S -> F

3) F -> F

4) S -> S

iij DSS

jij DSS

jiij DDSS

ij SS

Different Unions

a

d

c

b

a

b

c

d

a

b

d

c

Intersection Inclusive union Exclusive union

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

Example Instance:

Possible assignments T:

757565654321 ,,1,,,1,,,1,,,1,,,,,4 AAAAAAAAAAAAS

7654321 ,,,,,, AAAAAAAT 7654321 ,,,,,, AAAAAAAT

Resources

S (t)

CumulativeConsumption

Time

Start of theProject

End of theProject

R (t)

Time

Capacity curveC (t)

A0

A1

Non-renewable Renewable

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

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

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

Hierarchical Levels a) Progressive level

b) Regressive level

1)(max

1)( jpJ

Jip

iJji

i

1)(min

)(max)(

...1

jqL

jpLiq

iLji

Nji

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

Adjacency Matrix Aij

1 if there is a direct precedence i->j

0 if not

Level Adjacency Matrix Xij – number of

precendences links between level i and j

Example

2

3 4 5 10

9

876

1

Morphology and Simulation of Project Networks a) Series-network

b) Parallel-network

. . .

0

i=1 i=N

N+1

i=1

i=N

.

.

.

N

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

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

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

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

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

Distribution of tk in terms of I1 for the normal case

Distribution of tk in terms of I1 for the exponential case

Distribution of tk in terms of I2 for the normal case

Distribution of tk in terms of I2 for the Exponential case

Distribution of tk in terms of I4 for the normal case

Distribution of tk in terms of I4 for the exponential case

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...

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

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

Lower Bound Maximal value of all lower bounds

(super optima) Ignoring resources (CPM) Ignoring activities (for each resource):

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

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

Latest Starting Time, and AoN