Operations Research I:
Project Planning
Instructors: Martin Savelsbergh & Masoud Talebian
Project
There are a wide variety of activities to be completed, some of which can be performed concurrently but others that must be performed consecutively.
There are usually tight time lines and restricted budgets.
The timing and allocation of resources will affect the completion time.
For each activity there is a set of activities (called predecessors) that must be completed before the activity begins.
Example 1
Activity Immediate Predecessors Expected Time (Weeks)
A - 2
B A 3
C B 4
D B 6
E C 6
F C,D 7
G D 4
H E,F,G 5
Organized approach to accomplish the goal of
minimizing elapsed time of project defines objectives and activities
represents activities interactions on a network
estimates time and resources
Project Planning
Scheduling construction projects such as
office buildings, highways and swimming
pools
Developing countdown and “hold”
procedure for the launching of space crafts
Installing new computer systems
Designing and marketing new products
Completing corporate mergers
Building ships
Applications
Activity-on-Arc (AOA) Diagrams
Activities are represented by arcs and the nodes will be used to represent completion of a set of activities.
Time flows from left to right.
The event at the head of an arrow has a higher number than the event at the tail of the arrow.
All events, except the first and last, must have at least one activity arrow entering and one activity arrow leaving them.
No activity leaving an event can commence until all activities terminating from the event are complete.
1. Node 1 represents the start of the project. An arc should lead from node 1 to represent each activity that has no predecessors.
2. A node (called the finish node) representing the completion of the project should be included in the network.
3. Number the nodes in the network so that the node representing the completion time of an activity always has a larger number than the node representing the beginning of an activity.
4. An activity should not be represented by more than one arc in the network
5. Two nodes can be connected by at most one arc.
Activity-on-Arc (AOA) Diagrams
Dummy Activity
Imagine that a project involves a person going from A to B and dropping off some goods at B (activity “AB”) and then going on to C (activity “BC”). Meanwhile a second person goes from D to B and collects the goods (activity “DB”) and goes onto E (activity “BE”). Without a dummy activity the AOA diagram would look like:
BC
DB
1
3
5
8
9
AB
BE
Dummy Activity
Activity BE cannot commence until activities AB and DB have been completed. However activity BC can occur before DB is completed. This diagram does not reflect this so we introduce a dummy activity.
1
3
5
6
8
9 DB BE
BC AB
Example 1 AOA
activity duration
A,2 B,3
D,6
C,4 E,8
G,4
F,7 H,5
Activity-On-Node (AON) Diagrams Activities (or activities) are represented by nodes
Each activity has a duration denoted by dj
Node 0 represents the “start” and node n denotes the “finish” of the project
Precedence relations are shown by “arcs” specify what other activities must be completed before
the activity in question can begin.
form an acyclic graph (no directed cycles).
A path is a sequence of linked activities going from beginning to end
Example 1 AON
A,2 B,3 D,6
C,4
G,4
F,7
E,8
H,5
activity duration
Remark: the activities are labeled alphabetically.
If (i,j) is an arc then i < j.
Theorem. If a network has no directed cycles, then the
nodes can be labeled so that for each arc (i,j), i < j.
Such a node labeling is called a topological order.
1
2
5
4
3 6
7 8 5
1
3
2
1
1
2
4
7
3
6
With cycle 2-3-4-7-5-2, none of these 5 activities could be
scheduled first.
Project Network: acyclic graph
Managerial Roles (after Henry Mintzberg)
Interpersonal Figurehead
Leader
Liaison
Informational Roles Monitor
Disseminator
Spokesperson
Decisional Roles Entrepreneur
Resource Allocator
Disturbance Allocator
Negotiator
Development cycle:
Effort
Time
Specification Analysis Build Test Maintain
Alpha Beta
Estimation Techniques: Rules of Thumb Software projects:
estimate 10 x cost and 3 x time
1:3:10 rule 1: cost of prototype
3: cost of turning prototype into a product
10: cost of sales and marketing
>>Product costs are dominated by cost of sales
Hartree’s Law The time to completion of any project, as estimated by the
project leader, is a constant (Hartree’s constant) regardless of the state of the project
A project is 90% complete 90% of the time
80% Rule Don’t plan to use more than 80% of the available resources
Memory, disc, cycles, programming resource....
Project Control
Means of monitoring and revising the
progress of a project
Project Management Software
Explosive growth in software packages
using these techniques
Cost and capabilities vary greatly
Yearly survey in PM Network
Microsoft Project is most commonly used
package today
Free 60 day trial versions:
http://www.microsoft.com/office/98/project/
trial/info.htm
Critical Path Method (CPM)
A critical path for this project network
consists of a path from the start of the
project to the finish in which each arc in
the path corresponds to a constraint
having a non-negative dual price.
A-B-D-F-H is the longest path. Its length is 23
A,2 B,3 D,6
C,4
G,4
F,7
E,8
H,5
Theorem: The minimum length of the schedule is the
length of the longest path.
The longest path is called the critical path
Critical Path
Early Event Time (ET)
The earliest starting time for a particular activity is the earliest time at which we can start the activity assuming that all other activities take their estimated time to be completed.
The earliest finishing time for a activity is the earliest time at which the activity can be completed if all other activities are completed on time.
Early Event Time: Forward Pass
Find each immediate predecessor of node
i that is connected by an arc to node i.
To the ET for each immediate predecessor
of the node i, add the duration of the
activity connecting the immediate
predecessor to node i.
ET(i) equals the maximum of the sums
computed in previous step.
The Gantt Chart
0 2 4 6 8 10 12 14 16 18 20 22 24 26
A B
•Lay out the activities on a time line
•The width of the activity is the amount of time it
takes.
Example 1
A,2 B,3 D,6
C,4
G,4
F,7
E,8
H,5
Forward Pass
0 2 4 6 8 10 12 14 16 18 20 22 24 26
A B C
D
E
F
G
H
Late Event Time (LT)
The latest starting time is the latest time
that we can begin activity j without
delaying the overall completion of the
project.
The latest finishing time is the latest time
at which activity j can be finished without
delaying the project overall.
Find immediate successors of node i.
From the LT for each immediate successor to node i subtract the duration of the activity joining the successor the node i.
LT(i) is the smallest of the differences determined in previous step.
Late Event Time: Backward Pass
Example 1
A,2 B,3 D,6
C,4
G,4
F,7
E,8
H,5
Backward Pass
0 2 4 6 8 10 12 14 16 18 20 22 24 26
A B C
D
E
F
G
H
Example 1
The time to complete the schedule is at
least as long as any path.
A,2 B,3 D,6
C,4
G,4
F,7
E,8
H,5
The length of A-B-C-F-H is 21
For an arbitrary arc representing activity (i,j),
the total float, represented by TF(i,j), is the
amount by which the starting time of activity
(i,j), could be delayed beyond its earliest
possible starting time without delaying the
completion of the project.
Total float
Critical Path Method (CPM)
A set of critical activities that form a
continuous path from the beginning to the
end of the project is called a critical path
for the project.
0 2 4 6 8 10 12 14 16 18 20 22 24 26
C E A B
D F
G
H
0 2 4 6 8 10 12 14 16 18 20 22 24 26
A B C
D
E
F
G
H
Look for activities whose earliest start time and latest start time
are the same. These activities are critical, and are on a critical
path.
Critical Path Method
Advantage of ease of use
Lays out the Gantt chart (nicely visual)
Identifies the critical path
Used in practice on large projects
e.g., used for the big dig
Why CPM?
Incorporating Resource Constraints
Each activity can have resources that it
needs, like 3 construction workers, 1
crane, etc
In scheduling, do not use more resources
than are available at any time
Makes the problem much more difficult to
solve exactly. Heuristics are used.
Project Evaluation & Review
Technique (PERT)
CPM assumes that the duration of each
activity is known with certainty. For many
projects, this is clearly not applicable.
PERT is an attempt to correct this
shortcoming of CPM by modeling the
duration of each activity as a random
variable and incorporate variability in the
durations
Assume mean, m, and variance, s2, of the
durations can be estimated
PERT
Using the estimated mean durations, apply
the CPM calculation
To obtain an estimate of the variance of the
resulting makespan, add up the variances of
the activities on the critical path.
Since the distribution is assumed normal, we
can determine the probability that the
makespan of our project will not exceed an
upper bound.
PERT
The Central Limit Theorem (CLT)
nXXX ,,, 21
30n
nXXXX 21
X
)()( 1 nXEXEXE
)()()( 1 nXVXVXV
If
are independent random variables then for n
sufficiently large ( say) the random variable
may be approximated by a normal random variable
that has
and
PERT
For each activity in the project make three
estimations for the activity time:
a - an optimistic estimation , assuming that
everything goes well.
m - the most likely time for the completion of the
activity.
b - a pessimistic estimation assuming that
everything that can go wrong will go wrong.
Example 2
Activity Immediate
Predecessors
Activity time
Estimates
a m b
A - 3 5 7
B - 8 12.5 14
C - 5 7 9
D A,C 5 10.5 13
E B 1 4.5 5
F C 4 7.5 8
G D 2 11.25 13
H E,G,I 3 6.5 7
I F,D 8 12 16
J E,G,I 1 5.5 7
K N 1 2 3
L M 2 6.75 7
M H 1 2 3
N F 1 2 3
Let Tij be the duration of activity (i,j). PERT
requires the assumption that Tij follows a
beta distribution. According to this
assumption, it can be shown that the
mean and variance of Tij may be
approximated by
36
)(var
6
4)(
2abT
bmaTE
ij
ij
Example 2
“What is the probability that the project will
be completed in less than 48 time units?”
Crashing the Project Time
Crashing the Time
1. Perform a CPM (or PERT if appropriate) analysis of the project to determine a critical path (CP) for the project.
2. Calculate the cost per time unit (cptu) for each activity. List them in order of increasing cptu.
3. Select the activity with the smallest cptu. If this activity isn’t on the CP then we have finished step 3. If it is the CP then crash this activity by as much as possible except if The desired project time is reached. In this case reduce the activity
time as much as necessary and stop.
Reducing the activity time by this much changes the CP. In this case accelerate the activity up until the CP changes and then consider the new CP
4. Go back to step 3.
Example 3
Activity Immediate
Predecessor
NT
(days)
CT
(days)
NC
($1000’s)
CC
($1000’s)
A - 20 15 10 15
B A 10 5 12 16.5
C B 8 5 6 10.5
D A 11 10 4 5.5
E D,C 7 - - -
F E 6 - - -
G D 12 9 9 11
H E 13 10 12 16
I H,G 5 - - -
Example 3 How much will it cost to reduce the project time down
to 55 days?
How far can the project time be reduced?
Activit
y
Immediate
Predecessor
NT
(days)
CT
(days)
NC
($1000’s)
CC
($1000’s)
cptu
($1000/day)
A - 20 15 10 15 1
B A 10 5 12 16.5 0.9
C B 8 5 6 10.5 1.5
D A 11 10 4 5.5 1.5
E D,C 7 - - - -
F E 6 - - - -
G D 12 9 9 11 .6
H E 13 10 12 16 1.3
I H,G 5 - - - -
Example 3 AON
A,20 B,10
D,11
C,8
G,12
F,6
E,7 H,13 I,5
J
Example 3 Crashing Time
The project can be crashed to 55 days by
crashing B for 5 days and A for 3 days by
$7500 .
The project can be crashed at most to 47
days by crashing A, B, C, D, and H by
paying $18000.
Cost-Time Trade-Offs in PERT/CPM
time
cost
Suggested Reading
Sections 8.4 of
Operations Research: Applications and Algorithms, by W. Winston
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