the maximum profit is given by
`
(20× 3 + 30× 9) =
`
330.
We are now ready for the definition of linear programming
–
the technique of
solving the problem such as above.
What is
Linear Programming
Linear
because the equations and relationships introduced are linear. Note that
all the constraints and the objective functions are linear.
Programming
is used in
the sense of method, rather than in the computing sense.
B(3,9)
A(0,10)
‘Equal Profit’ lines
Direction of
Increasing Profit
10
0(0,0)

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82
Vectors and Three
Dimensional Geometry
In fact, linear programming is a technique for specifying how to use limited
resources or capacities of a business to obtain a particular objective, such as
least cost, highest margin or least time, when those resources have alternative
uses.
4.1
OBJECTIVES
After studying this unit, you should be able to:
define the terms-objective function,
constraints, feasible region,
feasible solution, optimal solution and linear programming;
draw feasible region and use it to obtain optimal solution;
tell when there are more than one optimal solutions; and
know when the
problem has no optimal solution.
4.2
LINEAR
PROGRAMMING
We begin by listing some definitions.
Definitions
*
In this unit we shall work with just two variables.
Objective Functions:
If
a
1
, a
2
, . . . , a
n
are constants and
x
1
x
2
, ..., x
n
are
variables, then the linear function Z
= a
1
x
1
+
a
2
x
2
+...+
a
n
x
n
which is to be
maximised or minimised is called objective function*.
Constraints:
These are the restrictions to be satisfied by the variables
x
1
,
x
2
..., x
n
.
These are usually expressed as inequations and equations.
Non-negative Restrictions:
The values of the variables
x
1
x
2
, ..., x
n
involved in the linear programming problem (LPP) are greater than or
equal to zero (This is so because most of the variable represent some
economic or physical variable.)
Feasible Region:
The common region determined by all the constraints of
an LPP is called the feasible region of the LPP.
Feasible Solution:
Every point that lies in the feasible region is called a
feasible solution. Note that each point in the feasible region satisfies all the
constraints for the LPP.
Optimal Solution:
A feasible solution that maximises or minimizes the
objective function is called an optimal solution of the LPP.

83
Linear Programming
A
Definition
The student may observe that a feasible region for a linear programming is a
convex region.
One of the properties of the convex region is that maximum and minimum values
of a function defined on convex regions occur at the corner points only. Since
all feasible regions are convex regions, maximum and minimum values of
optimal functions occur at the corner points of the feasible region.
The region of Figure 3 is convex but that of Figure 4 is not convex.
y
y
o
x
o
x
Figure 3
: Convex Region
Figure 4 : Not Convex
4.3
TECHNIQUES OF SOLVING LINEAR PROGRAMMING
PROBLEM
There are two techniques of solving an L.P.P. (Linear Programming Problem)
by graphical method.
These are
(i)
Corner point method, and
(ii)
Iso profit or Iso-cost method.

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- Spring '17