How to solve Linear Programming Problem

In this session let me help you go through basically on Problems on Linear Programming problems. With the following examples you will come to know how to solve LPP.

Problem 1
Solve the following linear programming problem graphically:
Maximize Z = 4x + y ... (1)
subject to the constraints:
x + y ≤ 50 ....(2)
3x + y ≤ 90 ... (3)
x ≥ 0, y ≥ 0 ... (4)

Solution -
The shaded region in the below figure helps to understand graphical feasible region determined by the system of constraints (2) to (4). We observe that the feasible region OABC is bounded. So,
we now use Corner Point Method to determine the maximum value of Z.
The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and
(0, 50) respectively. Now we evaluate Z at each corner point. This will also help us in factoring calculator.

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Corner Point Method in Linear Progamming Problems

Corner Point Method is an important one to solve linear Programming Problems.

Following are the method comprises of the following steps:

(i) Find the feasible region of the linear programming problem and determine
its corner points (vertices).
(ii) Evaluate the objective function Z = ax + by at each corner point. Let M
and m respectively be the largest and smallest values at these points.
(iii) If the feasible region is bounded, M and m respectively are the maximum
and minimum values of the objective function.

If the feasible region is unbounded, then

(i) M is the maximum value of the objective function, if the open half plane
determined by ax + by > M has no point in common with the feasible
region. Otherwise, the objective function has no maximum value.
(ii) m is the minimum value of the objective function, if the open half plane
determined by ax + by < m has no point in common with the feasible
region. Otherwise, the objective function has no minimum value.