LINEAR PROGRAMMING PROBLEM
INTRODUCTION
In a decision-making embroilment, model formulation is important because it represents the essence of business decision problem. The term formulation is used to mean the process of converting the verbal description and numerical data into mathematical expressions which represents the relevant relationship among decision factors, objectives and restrictions on the use of resources. Linear Programming (LP) is a particular type of technique used for economic allocation of 'scarce' or 'limited' resources, such as labour, material, machine, time, warehouse space, capital, energy, etc. to several competing activities, such as products, services, jobs, new equipment, projects, etc. on the basis of a given criterion of optimally. The phrase scarce resources means resources that are not in unlimited in availability during the planning period. The criterion of optimality, generally is either performance, return on investment, profit, cost, utilily, time, distance, etc.
George B Dantzing while working with US Air Force during World War II, developed this technique, primarily for solving military logistics problems. But now, it is being used extensively in all functional areas of management, hospitals, airlines, agriculture, military operations, oil refining, education, energy planning, pollution control, transportation planning and scheduling, research and development, etc. Even though these applications are diverse, all I.P models consist of certain common properties and assumptions. Before applying linear programming to a real-life decision problem, the decision-maker must be aware of all these properties and assumptions, which are discussed later in this chapter.
Before discussing in detail the basic concepts and applications of linear programming, let us be clear about the two words, linear and programming. The word linear refers to linear relationship among variables in a model. Thus, a given change in one variable will always cause a resulting proportional change in another variable. For example, doubling the investment on a certain project will exactly double the rate of return. The word programming refers to modelling and solving a problem mathematically that involves the economic allocation of limited resources by choosing a particular course of action or strategy among various alternative strategies to achieve the desired objective.
A large number of computer packages are available for solving a mathematical LP model but there is no general package for building a model. Model building is an art that improves with practice. To illustrate, how to build I.P models, a variety of examples are given in this chapter.
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