__Linear Programming Problem__**Definition**

Linear programming is a technique for
determining an optimum schedule of independent activities in view of the
available resources. OR,

Linear programming is the maximization or minimization of a linear function of variables subject to constraint of linear inequalities. Linear programming is a mathematical technique designed to assist the organization in optimally allocating it’s available resources under conditions of certainty in problems of scheduling product mix and so on.

Linear programming is the maximization or minimization of a linear function of variables subject to constraint of linear inequalities. Linear programming is a mathematical technique designed to assist the organization in optimally allocating it’s available resources under conditions of certainty in problems of scheduling product mix and so on.

**Problem of Constrained Optimization:**

Many businesses and economic situations
are concerned with a problem of planning activity. Such as-

i.
Which
commodity/commodities should be produced?

ii.
What
quantities of commodity should be produced?

iii.
By which
process the commodity should be produced?

In each case, there are limited resources
at their disposal and their problem is to make such a use of these resources so
as to yield the maximum production or to minimize the cost of production, or to
give the maximum profit, etc. such problems are referred to as the problems of
constrained optimization.

**Linear programming:**

Linear programming is a mathematical
technique for determining the optimum allocation of resources and obtaining a
particular objective when there are alternative uses of resources such as
money, manpower, material, machine and other facilities.

The objective in allocation of resource
may be cost minimization or inversely profit maximization. The term linear
implies that all the mathematical relations used in the problems are linear
relation, while the term programming refers to the method of determining a
particular program or plan of action.

**Meaning of Linear Programming:**

Linear programming is a mathematical way
of planning, which involves three steps:

a. Identify the objective function as a
linear function of its variables and state all the limitations on resources as
linear equations and or in equations (constraints).

b. Use mathematical techniques to find all possible
sets of values of the variables (unknowns), satisfying the constraints.

c. Select the particular set of variables
obtained in (2) that lead to the objective such as maximum profit or minimum
cost.

The result at step (1) above is called a
linear programming problem. The set of solutions obtained in (2) is called the
set of feasible solutions and the solution finally selected in step (3) is
known as the optimum or optima or the best solution of the linear programming
problem.