Symbiosis Institute of Management Studies (SIMS)

Project Management

PERT,CPM, Resource Allocation and GERT

Manuja Goenka, E-12Raj Jyoti Das-E-13

August 2013

• Two commonly used network methods for planning and scheduling are:

1. Program Evaluation and Review Techniques (PERT)

2. Critical path Method (CPM)

*Both PERT & CPM are termed critical path methods

Tools for Scheduling

History of PERT

• Project Evaluation and Review Technique (PERT)

– U S Navy (1958) for the POLARIS missile program– Multiple task time estimates (probabilistic nature)

PERT • PERT is based on the assumption that an activity’s

duration follows a probability distribution instead of being a single value

• Three time estimates are required to compute the parameters of an activity’s duration distribution:– pessimistic time (a) - the time the activity would take

if things did not go well– most likely time (m ) - the consensus best estimate of

the activity’s duration– optimistic time (b) - the time the activity would take if

things did go wellMean (expected time):te =

a + 4m + b6

Variance: V =b- a

6

2

Three Time Estimates of PERT

•Optimistic

•Most Likely

•Pessimistic

PERT uses three time estimates to address uncertainty of project duration-

It is the time where there is 50-50 chances that the activity will be completed earlier or later than it.

For this case :

Mean or expected Time

Variance is measure of variability in the activity completion period.

Variance

The larger V, the less reliable te

The Expected Duration Of The Project

The expected duration of the project (Te) is the sum of the expected activity times along the critical path

Te = ∑ te

Where te are expected times of the activities on the critical path

The variation in the project duration distribution is computed as the sum of the variances of the activity durations along the critical path:

Vp = ∑ V

Where V is the variance of critical path

The Variation In The Project Duration

Near Critical Path

Path (events)

Te = ∑ te Vp = ∑ V

1-2-6-8 28** 6.34

1-7-8 20 17.00

1-2-5-7-8 Te=29* Vp=6.00

1-4-5-7-8 18 3.89

1-3-4-5-7-8

27** 12.00

darla/smbs/vit 11

29 Time

Z

Probability

27

0.207

Let us assume,Expected completion duration of a project = 29 weeksVariance of the project duration = 6Then what will be the probability of finishing the project by 27 weeks can be calculated by the formula:

Therefore,Z=(27-29)/2.449 =-0.82Prob. Of finishing theproject by 27 weeksis app. 21%

Probability of Finishing a Project

History of Critical Path Method (CPM)

• E I Du Pont de Nemours & Co. (1957) for construction of new chemical plant and maintenance shut-down

• CPM is a “Deterministic” approach• CPM includes mathematical procedure for estimating

the trade off between project duration and project cost

• CPM emphasis on applying additional resources to particular key activities

Normal Time, Tn: It is the time taken by an activity under normal work conditionsNormal Cost, Cn: The cost incurred in doing an activity in normal time.

Time-Cost Relationship

CrashingAn activity is said to be crashed when maximum effort is applied to finish that activity in the shortest possible time.

Cost Slope

• The cost slope shows by how much the cost of job would change if activities were speed up or slowed down.

In this case,Cost Slope= (18-9)/(5-8) = $3 K/week

CPM calculation• Path

– A connected sequence of activities leading from the starting event to the ending event

• Critical Path– The longest path (time); determines the project

duration• Critical Activities

– All of the activities that make up the critical path

Forward Pass• Earliest Start Time (ES)

– earliest time an activity can start – ES = maximum EF of immediate predecessors

• Earliest finish time (EF)– earliest time an activity can finish– earliest start time plus activity time (EF= ES + t)

Latest Start Time (LS)Latest time an activity can start without delaying critical path time

LS= LF - tLatest finish time (LF)

latest time an activity can be completed without delaying critical path timeLS = minimum LS of immediate predecessors

Backward Pass

CPM calculation

Project Crashing• Crashing

– reducing project time by expending additional resources• Crash time

– an amount of time an activity is reduced• Crash cost

– cost of reducing activity time• Goal

– reduce project duration at minimum cost

Time-Cost Relationship Crashing costs increase as project duration decreases Indirect costs increase as project duration increases Reduce project length as long as crashing costs are less than

indirect costsTime-Cost Tradeoff

cost

time

Direct cost

Indirect cost

Total project cost

A

D

G

FC

BE

98

87

5

5

5

022

149

178

22

10

0

917

17

Activity

Normal(Wks)

Crush(Wks)

Cost Slope(K$)

Tn Cn Tc Cc

A 9 10 6 16 2

B 8 9 5 18 3

C 5 7 4 8 1

D 8 9 6 19 5

E 7 7 3 15 2

F 5 5 5 5 -

G 5 8 2 23 5

Crashing Example

A

D

G

FC

BE

98

87

5

5

5

022

149

178

22

10

0

917

17

Critical Path A-D-G=22wks

Types of Project Constraints

• Technical or Logic Constraints– Constraints related to the networked sequence in

which project activities must occur.• Physical Constraints

– Activities that cannot occur in parallel or are affected by contractual or environmental conditions.

• Resource Constraints– The absence, shortage, or unique interrelationship and

interaction characteristics of resources that require a particular sequencing of project activities.

The Resource Problem

• Resources and Priorities– Project network times are not a schedule until resources

have been assigned.• The implicit assumption is that resources will be available in

the required amounts when needed.• Adding new projects requires making realistic judgments of

resource availability and project durations.

• Resource-Constrained Scheduling– Resource leveling (or smoothing) involves attempting to

even out demands on resources by using slack (delaying noncritical activities) to manage resource utilization.

Kinds of Resource Constraints

• People

• Materials

• Equipment

• Working Capital

Classification of A Scheduling Problem

• Time Constrained Project– A project that must be completed by an imposed

date.• Time is fixed, resources are flexible: additional resources

are required to ensure project meets schedule.

• Resource Constrained Project– A project in which the level of resources available

cannot be exceeded.• Resources are fixed, time is flexible: inadequate resources

will delay the project.

27

Example :

• Without resource constraints relatively easy

• With resource constraints very complex:when jobs share resources with limited availability, these jobs cannot be processed simultaneously

Jobs 1 2 3 4 5p(j) 8 4 6 4 4

R(1,j) 2 1 3 1 2R(2,j) 3 0 4 0 3

Resource R1 R2Available 4 8

1 4

2 5

3

28

'jj

''jj

''j

'j

'j

SpCslack

j job of time completion possiblelatest C

j job of time completion possibleearliest C

j job of time starting possibleearliest S

29

Resource constraints

• Suppose jobs require a resource:

resource requirements321 4 5 6 7 8

3

2

1

4

5

6

1

2

3

4

5

6

Job p(j) Predecessors S' C'' R(1,j)1 2 - 0 3 32 3 - 0 3 13 1 - 0 6 24 4 1,2 3 7 25 2 2,3 3 8 36 1 4 7 8 3

30

Resource constraints (cont.)• Suppose :

Cmax increases by 2321 4 5 6 7 8

3

2

1

4

5

6

4R1

9 10

1

2

3 45 6

31

Resource-Constrained Project Scheduling Problem (RCPSP)

• n jobs j=1,…,n• N resources i=1,…,N• Rk: availability of resource k

• pj: duration of job j

• Rkj:requirement of resource k for job j

• Pj: (immediate) predecessors of job j

32

RCPSP

• Goal: minimize makespan:• Restrictions:

– no job may start before T=0– precedence relations– finite resource capacity

'j

jmax CmaxC

33

RCPSP example

4R1

0 2 4 6 8 10

2

3 4 562

4

2R2 0 2 4 6 8 10

1 23

4 56

2

12

Job p(j) P(j) S' C'' R(1,j) R(2,j)1 2 - 0 3 3 22 3 - 0 3 1 13 1 - 0 6 2 14 4 1,2 3 7 2 15 2 2,3 3 8 3 26 1 4 7 8 3 1

1

Loading And Leveling

• Loading- amount of a resource necessary to conduct a project– Depends on the requirements of individual

activities.– Changes throughout a project

• Resource Leveling- process of scheduling activities so that the amount of a certain required resource is balanced throughout the resource.

Multiproject Resource Schedules

• Multiproject Scheduling Problems– Overall project slippage

• Delay on one project create delays for other projects

– Inefficient resource application• The peaks and valleys of resource demands create

scheduling problems and delays for projects.

– Resource bottlenecks• Shortages of critical resources required for multiple

projects cause delays and schedule extensions.

Multiproject Resource Schedules

• Managing Multiproject Scheduling– Create project offices or departments to oversee

the scheduling of resources across projects.

– Use a project priority queuing system: first come, first served for resources.

– Centralize project management: treat all projects as a part of a “megaproject.”

– Outsource projects to reduce the number of projects handled internally.

Limitations of PERT/CPM• All immediate predecessor activities must be

completed before a given activity can be started.• No activity can be repeated and no “looping back”• Duration time for an activity is restricted to Beta

Distribution PERT and a single estimate in CPM.• Critical Path is always considered the longest.• There is only one terminal event and the only way

to reach it is by completing all activities in the project

GERT

• A network analysis technique used in project management.

• It allows probabilistic treatment of both network logic and activity duration estimated.

• The technique was first described in 1966 by Dr. Alan B. Pritsker of Purdue University and WW Happ.

• Compared to other techniques, GERT is an only rarely used scheduling technique.

Contd..

• Utilizes probabilistic and branching nodes• It represents the node will be reached if any m

of its p immediate predecessors are completed.

m

n

p

Contd..• It represents a probabilistic output where any

of q outputs are possible• Each branch has an assigned probability• When no probability is given, the probability is

assumed to be one for each branch.1

2

q

Example

 

Thank YOu