Operational Research I

By Dr. SW Poon

Definitions O. R. Society of Great Britain

“OR is the application of the methods of science to complex problems arising in the direction and management of large systems of men, machines, materials and money in industry, business, government and defence”

O. R. Society of America “OR is concerned with scientifically deciding how to

best design and operate man-machine systems, usually requiring the allocation of scarce resources”

Models in O.R.

When does a problem exist? Daellenbach and George:

There must be a decision-maker who must have an objective

The decision-maker must have at least two alternative courses of action available

There must be some degree of doubt as to which course of action best achieves the objective

There must be an environment to which the problem pertains (firm, economy)

Models in O.R. (Cont’) A mathematical model consists of decision variables

linked by constraining relationships ad entering into an objective function. In the model, there will be certain fixed factors (constants, parameters, coefficients) which are data. There may be uncontrollable factors such as random variables

Yet, the model is UNREALISTIC as it fails to reflect the important factors and relationships in a situation. The purpose of building a model is: To identify the best values of the decision variables To study the system so as to find ways to improve its

performance

Steps in an operational research project

Network Diagrams

The traditional method Network diagrams

Producing an Activity List Drawing the Network Diagrams Dummy Activities

Six steps in the Critical Path Method

Step 1: Identification of project activities Step 2: Network formulation Step 3: Duration estimation Step 4: Forward pass computation Step 5: Backward pass computation Step 6: Tracing the critical path

The float of an activity Float = latest finish time of an activity – earliest finish time of the activity Independent float: the time by which an activity can

expand without affecting any other activity, either previous or subsequent

Free float: the time by which an activity can expand without affecting subsequent activities. If it is absorbed at the planning stage the float in earlier activities will be reduced. Once a project is under way, the free float in an activity can be used once the tail event is reached without affecting any other activity in the network

The float of an activity (Cont’)

Total float: the time by which an activity can expand. When total float is absorbed at the planning stage, the floats in both previous and subsequent activities may be reduced.

Interfering float: the difference between total and free float (The use of this float reduces float available to subsequent activities. If INTF is fully used then subsequent activities become critical.

What is PERT?

PERT (Programme Evaluation and Review Technique) uses three quantities in estimating the duration of a single activity. They are: The optimistic time The pessimistic time The most likely time

The Probability Concept of PERT

The Probability Concept of PERT (Cont’)

Preference Network Diagram No dummy activity is necessary in a precedence network A precedence network is usually simpler than an AOA

network for the same number of activities, because there are no dummies;

In a precedence network, each activity can be assigned a single unique number to represent it; but in an AOA network, an activity has to be represented by two numbers, i and j. Hence, the creation and modification of a precedence network diagram is easier; and

Less arithmetic will be involved in the forward pass and backward pass in a precedence network since there are no dummies

Introduction to transportation problem The general form of a transportation problem A special algorithm

Example 1: Transportation problem with Demand equal to supply

Example 1 (Cont’)

Example 2: Transportation problem with demand not equal to supply

The MODI method: despatching cost and receiving cost

The general form of an assignment problem

Using the MODI method to solve an assignment problem