Definition of Linear Programming

            Linear programming is a mathematical technique for determining the optimal allocation of resources and obtaining a particular objective when there are alternative uses of the resources: money manpower material, machine and other facilities. The objective in resource allocation may be cost minimization or inversely profit maximization. The technique of linear programming is applicable to problems in which the total effectiveness can be expressed as a linear function of individual allocations and limitations on resources give rise to linear equalities or inequalities of the individual allocations.

            According to kohlar " A method of  planning and operation involved in the construction of a model of a real situation containing the following element:

(a)    Variables representing the available choices and (b) mathematical expressions (i) relating the variables to the controlling conditions and (ii) reflecting the criteria to be used in measuring the benefits derivable from each of the several possible plans and (iii) establishing the objective. The method may be so devised as to ensure the selection to the best of a large number of alternatives."

Samuelson Dorfman and Solow define LP as "The analysis of problems in which a linear function of a number of variables is to be restraints in the form of linear inequalities."

            In the words of Loomba "LP is only one aspect of what has been called a system approach to management where in all programmers are designed and evaluated in terms of their ultimate affects in the realization of business objectives."

Remark. Linear programming is one of the most widely used and best understood Operations Research techniques. The LP is concerned with the problem of allocation limited resources among competition activities in an optimal manner. The allocation process involves assigning resources, (e,g., man hours money, machine hours raw material etc.)  to specific activities in such a way that the defined objectives are optimized. As there is only a limited supply of most of productive resources, they must be judiciously assigned subject to specified restraining conditions. For example:

1                    The production manager wants to allocate the available machine time and labour  hours in each department along with the raw material to the activities of producing the different products which have been scheduled. The manager would like to like to determine the number of units of each of the product to be produced so as to maximize the profit.

2                    A manufacturer wants to develop a production schedule and an inventory policy that will satisfy sales demand in future periods. Ideally the schedule and policy will enable company to satisfy demand and at the same time minimize the total production and inventory costs.

3                    A financial analyst must selct an investment portfolio from a variety of stock and bobd investment alternatives. The analyst would like to establish the portfolio that maximizes the return on investment.

4                    A marketing manager wants to determine how he should allocated a fixed advertising budget among alternative advertising media such as radio television, newspaper and magazines. The manager would like to determine the media schedule that maximizes the advertising effectiveness.

5                    A company has warehouses in a number of locations throughout the country that are intended to serve its many markets. Given a set of customer demands for its products, company would like to determine which warehouse ship should show much product to which market so that the total transportation costs are minimized.

Althouth these are but a few of the possible situations where linear programming has been used successfully, the examples do point out the broad nature of the types of problems that can be tackled using linear programming. Even though the applications are diverse a close scrutiny of the examples points out one basic property that all these problems have in common in each example we were concerned with maximizing or minimizing some quantity. In example 2 we wanted to minimize costs 3 we wanted to maximize return on investment in example 4 we wanted to maximize total advertising effectiveness, and example 5 we wanted to minimize total transportation costs. In linear programming terminology the maximization or minimization of a quantity is referred to as the objective of the problem. Thus the objective of all linear programmers is to maximize or minimize some quantity.

Second property common to all linear programming problems is that there are restrictions or constraints that limit the degree to which we can pursue our objective.

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