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Introduction 1
Network is a technique used for planning and scheduling of large projects in the fields of construction, maintenance, fabrication, purchasing, computer system instantiation, research and development planning etc. There is multitude of operations research situations that can be modeled and solved as network. Some recent surveys reports that as much as 70% of the real-world mathematical programming problems can be represented by network related models. Network analysis is known by many names _PERT (Programme Evaluation and Review Technique), CPM (Critical Path Method), PEP (Programme Evaluation Procedure), LCES (Least Cost Estimating and Scheduling), SCANS (Scheduling and Control by Automated Network System), etc
This chapter will present three of algorithms.
1. PERT & CPM
- Shortest- route algorithms
- Maximum-flow algorithms
The Basic Terminology
Network
It is a graphical representation of logical and sequentially connected activities and events of
a project. Network is also called arrow diagram. PERT (Programme Evolution Review
Technique) and (Critical Path Method) are the two most widely applied techniques.
Project
A project is defined as a combination of interrelated activities which must be executed in a
certain order in for its completion.
Project Management Process
Network analysis is the general name given to certain specific techniques which can be used
for the planning, management and control of projects
Activity
Any individual operation, which utilizes resources and has an end and a beginning, is called activity. A task or a certain amount of work required in the project Requires time to complete Represented by an arrow
These are usually classified into four categories: Predecessor activity Successor activity
Network Techniques
PERT CPM
Concurrent activity Dummy activity
Dummy Activity It Indicates only precedence relationships and does not require any time of effort
PERT(Program Evaluation and Review Technique) is a method to analyze the involved
tasks in completing a given project, especially the time needed to complete each task, and
identifying the minimum time needed to complete the total project.
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 (tp ) - the time the activity would take if things did not go well
- Most likely time (tm ) - the consensus best estimate of the activity’s duration
- Optimistic time (to ) - the time the activity would take if things did go well.
Mean (expected time) =
( tp 4 tm to )
Variance ( 2 ) =
2
tp t o
Probability computation: Determine probability that project is completed within specified
time
X
Z
Where = project mean time = project standard mean time x = (proposed) specified time
Float: Float of an activity represents the excess of available time over its duration.
Total Float (Ft) The amount of time by which the completion of an activity could be delay beyond the earliest expected completion time without affecting the overall project duration. i.e. Tf= (Latest start-Earliest start) for activity(i-j), or,(Tf)ij=(LS)jj-(ES)ij
Free Float (Ff) The time by which the completion of an activity can be delayed beyond the earliest finish time without affecting the earliest start of a subsequent (succeeding) activities.
Situations in network diagram
- A must finish before either B or C can start
Distinguish Between PERT and CPM? WBUT-08 4
PERT
(Programme Evaluation Review Technique)
CPM
(Critical Path Method)
- PERT is event oriented.
- PERT is probabilistic.
- PERT is primarily concerned with time only.
- PERT is generally used for projects where time required to complete the activities is not known a priori. Thus PERT is used for large, R&D type of projects.
- Three time estimates are possible for activities linking up two events.
- CPM is activity oriented.
- CPM is deterministic.
- CPM places dual emphasis on project time as well cost.
- CPM is used for projects which are repetitive in nature and comparatively small in size. 5.One time estimate is possible for activities (No allowance is made for uncertainty)
Example: 02. The following table gives the activities of a construction project and other relevant information.
Activities (i-j)
Normal duration (days)
Crash duration (days)
Crashing cost (Rs. per day) 1- 1- 1- 2- 3- 4-
A. What is the normal project length and minimum project length? B. Determine the minimum crashing costs of schedule ranging from length down to and the minimum length schedule. C. What is the optimal length schedule duration of each job for your solution? Given that over head cost total Rs. 60 per day.
E2 = 9,
L2=13,
E2=20,
L2=20,
E2=8,
L2=8,
E2=18,
L2=18,
E1=0,
L1=0,
Solution:
A. The Critical path is 1 3 4 5 with normal duration 20 days and minimum project
length is 12 days.
Normal Project
length(days)
Crashing Cost (day/Rs.) Overhead cost @
Rs. 60/day
Total Cost(Rs.)
B. Total cost increasing for 14 days duration, the minimum total cost Rs. 1030 occurs 15
days duration.
C. Optimum duration of each job is as follows:
D.
Job: (1,2) (1,3) (1,4) (2,4) (3,4) (4,5)
Optimum duration days: 9 8 14 4 6 1
Example-3: The time estimates (in hours) for the activities of a PERT network are given below:
Activity t 0 tm tp 1-2 1 1 7 1-3 1 4 7 1-4 2 2 8 2-5 1 1 1
b) Determination of project paths
Length of the path 1→ 2→5→6 = 2+1+7=
Length of the path 1→ 3→5→6 = 4+6+7=
Length of the path 1→ 4 →6 = 5+3= 8
Since 1→ 3→5→6 has largest duration. Therefore the critical path is1→ 3→5→6.
c) The expected project length duration is = 17 weeks
d) Standard deviation of project length (σ^2 )=sum of the standard deviations of the activities on
the critical paths=1+4+4 =
e) I) the probability of completing the project with in 4 weeks earlier than expected is given by
P ( Z D ), where D TS^ TE
D. Given that P ( Z 1. 33 ) 0. 9082. Therefore
P ( Z 1. 33 ) 0. 5 ( 1. 33 )
ii) The probability of completing the project not more than 4 weeks than the expected time
is given by S^ E
T T
P ( Z D ), where D
D. Given that P ( Z 1. 33 ) 0. 9082. Therefore
P ( Z 1. 33 ) 0. 5 ( 1. 33 )
f) Value of Z for p=90% i.e. 1. 28
TS 17
, Therefore TS=1.28×3+17=20.84 weeks
3
Example-06: Find the shortest path between every two nodes by Floyd’s algorithms?
Solution: Using Floyd’s algorithms, we have
Iteration-0, k=
D 0 1 2 3 4 5
1 — 3 10
2 3 —
3 10 — 6 15
4 5 6 — 4
5 4 —
S 0
Iteration-1,k=
D 1
1 2 3 4 5
1 — 3 10
2 3 — 13 5
3 10 15 — 6 15
4 5 6 — 4
5 4 —
S 1
Therefore Maximum flow in the network is F=f1+f2+f3+f4+f5=20+10+10+10+10=60 units.
Exercises
- Distinguish between PERT and CPM. What is Critical path? WBUT-
- Construct a network for each of the activities and their precedence relationships are given below:
Activities A B C D E F G H I J K Predecessor - - A A I,J,K B,D B,D F A G,H F
- Find the shortest routes between city 1 to city 2 by Dijkstra’s algorithm.
- Find the maximum flow in the network
- Draw a network diagram for the following project & find critical path and critical activities and project duration. WBUT- 08
Activity Predecessor Duration (days) A --- 6 B A 4 C B 7 D A 2 E D 4 F E 10 G --- 2
