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