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Operations Research

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Operations Research (MTH601)
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Ignizio, J.P., Linear Programming in Single and Multiple Objective System. Prentice-Hall
Segment II: PERT / CPM
Lectures 4 -10
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Operations Research (MTH601)
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INTRODUCTION
PERT is an acronym for "Programme Evaluation and Review Technique". This was created as a means to
plan and accelerate the development of the Polaris Ballistic Missile. In USA the defence department developed a
nuclear missile to be launched from beneath the ocean's surface by a mobile submarine, which would be an effective
deterrent against aggression by an enemy. This paved way to plan how to design, develop and plan the different
stages in the production of a missile and how quickly this task could be completed. A planning and scheduling
technique named PERT gave the answer to these questions.
In any new venture, uncertainties are bound to creep in. PERT incorporated these uncertainties into a
model, which provides a reasonable answer to these uncertainties. There are certain statistical aspects scheduling
large projects consisting of numerous activities whose completion times are uncertain and are independent of one
another. PERT is an event-oriented technique. By 'event' we mean reaching a certain stage of completion of the
project.
Another technique, Critical Path Method, abbreviated as CPM, has emerged simultaneously. It is also a
network technique but it is concerned with obtaining the trade-off, between cost and completion date for large
projects. In any project consisting of several activities each activity can be completed in a normal duration with
normal cost. If we employ more persons or skilled people or given overtime to the workers, the activity could be
completed in a reduced duration known as crash duration. But this involves an increased cost in the form of
additional resources. With CPM the amount needed to complete the various activities is assumed to be known with
certainty. So, the direct costs for the activities increase and hence the cost of the project also increases. By reducing
the activity duration of some or all possible completed ahead of the schedule. This will naturally reduce the
overhead cost for the entire project. On one hand the direct expenses increase, if we shorten the activity duration,
but, the indirect expenses for the project are reduced. We have to strike a balance or an optimum time schedule, or a
least cost schedule is to be obtained. This is the purpose of the Critical Path Method. Thus CPM is not concerned
with uncertain job times as in PERT. PERT is useful in research and developmental projects, whereas CPM is
mostly used in construction projects, or in situations already handled, so that the details like the normal completion
time, crash duration and cost of crashing are already known.
The following are the suggested applications when PERT or CPM is found useful.
The construction of a building or of a highway.
Planning and launching a new product.
Scheduling maintenance for a project.
The manufacture and assembly of a large machine tool.
To conduct a music or drama festival.
Preparation of budget for a company.
CONCEPT OF NETWORK
The first step in the application of CPM / PERT is to develop a network representation of the project plan.
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A 'network' is defined as a graphic representation with a flow of some type in its branches. It represents
nodes and branches. Below in table 1, we represent different systems satisfying the definition of network in the
physical world.
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Operations Research (MTH601)
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Table 5.1
Physical
Nodes
Branches
Flow
situation
Highway systems
Intersections
Roads
Vehicles
Communication
Switching points
Wires
Messages
systems
Fluid supply
Pumping stations
Pipes
Fluid
systems
Production
Work centres
Handling routes
Jobs
systems
Project
Decision points
Activities
Time
Management
Airway systems
Airports
Airlines
Aircraft
A node is the intersection of the two branch lines. It is denoted by a circle. Each branch represents an
activity. Each node represents an event, which is a specific definable accomplishment recognizable at a particular
instant of time. The arrowheads indicate the sequence in which events must be achieved. Thus an event is the
completion of all the activities leading into that node and this event must precede the initiation of the activity leading
out of the node.
1
2
node
Branch
node
Fig. 1
An arrow diagram represents a project graph. An arrow connecting two nodes, representing two events,
represents each activity. The head of the arrow identifies the start of the activity.
Let us take an example and illustrate the construction of arrow diagram for a project.
Example 1: A company is interested in preparing a budget. The details of the activities and the departments
involved are given in the Table 2.
Table 2
The project of preparation of a Production budget.
Job
Alternate
Job
Department
Identification
Description
A
1-2
Forecasting sales
Sales
B
2-4
Pricing Sales
Sales
C
2-3
Preparing production Schedule
Engineering
D
3-4
Costing the production
Costing
E
4-5
Preparation of budget
President
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Table of Contents:
  1. Introduction:OR APPROACH TO PROBLEM SOLVING, Observation
  2. Introduction:Model Solution, Implementation of Results
  3. Introduction:USES OF OPERATIONS RESEARCH, Marketing, Personnel
  4. PERT / CPM:CONCEPT OF NETWORK, RULES FOR CONSTRUCTION OF NETWORK
  5. PERT / CPM:DUMMY ACTIVITIES, TO FIND THE CRITICAL PATH
  6. PERT / CPM:ALGORITHM FOR CRITICAL PATH, Free Slack
  7. PERT / CPM:Expected length of a critical path, Expected time and Critical path
  8. PERT / CPM:Expected time and Critical path
  9. PERT / CPM:RESOURCE SCHEDULING IN NETWORK
  10. PERT / CPM:Exercises
  11. Inventory Control:INVENTORY COSTS, INVENTORY MODELS (E.O.Q. MODELS)
  12. Inventory Control:Purchasing model with shortages
  13. Inventory Control:Manufacturing model with no shortages
  14. Inventory Control:Manufacturing model with shortages
  15. Inventory Control:ORDER QUANTITY WITH PRICE-BREAK
  16. Inventory Control:SOME DEFINITIONS, Computation of Safety Stock
  17. Linear Programming:Formulation of the Linear Programming Problem
  18. Linear Programming:Formulation of the Linear Programming Problem, Decision Variables
  19. Linear Programming:Model Constraints, Ingredients Mixing
  20. Linear Programming:VITAMIN CONTRIBUTION, Decision Variables
  21. Linear Programming:LINEAR PROGRAMMING PROBLEM
  22. Linear Programming:LIMITATIONS OF LINEAR PROGRAMMING
  23. Linear Programming:SOLUTION TO LINEAR PROGRAMMING PROBLEMS
  24. Linear Programming:SIMPLEX METHOD, Simplex Procedure
  25. Linear Programming:PRESENTATION IN TABULAR FORM - (SIMPLEX TABLE)
  26. Linear Programming:ARTIFICIAL VARIABLE TECHNIQUE
  27. Linear Programming:The Two Phase Method, First Iteration
  28. Linear Programming:VARIANTS OF THE SIMPLEX METHOD
  29. Linear Programming:Tie for the Leaving Basic Variable (Degeneracy)
  30. Linear Programming:Multiple or Alternative optimal Solutions
  31. Transportation Problems:TRANSPORTATION MODEL, Distribution centers
  32. Transportation Problems:FINDING AN INITIAL BASIC FEASIBLE SOLUTION
  33. Transportation Problems:MOVING TOWARDS OPTIMALITY
  34. Transportation Problems:DEGENERACY, Destination
  35. Transportation Problems:REVIEW QUESTIONS
  36. Assignment Problems:MATHEMATICAL FORMULATION OF THE PROBLEM
  37. Assignment Problems:SOLUTION OF AN ASSIGNMENT PROBLEM
  38. Queuing Theory:DEFINITION OF TERMS IN QUEUEING MODEL
  39. Queuing Theory:SINGLE-CHANNEL INFINITE-POPULATION MODEL
  40. Replacement Models:REPLACEMENT OF ITEMS WITH GRADUAL DETERIORATION
  41. Replacement Models:ITEMS DETERIORATING WITH TIME VALUE OF MONEY
  42. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS
  43. Dynamic Programming:Analysis of the Result, One Stage Problem
  44. Miscellaneous:SEQUENCING, PROCESSING n JOBS THROUGH TWO MACHINES
  45. Miscellaneous:METHODS OF INTEGER PROGRAMMING SOLUTION