Linear Programming Problems

Solving linear Programming Problems with the help of graph

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.

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This is how you understand how to solve Linear Programming Problems

Graphical method of Solving Linear Programming

What is Linear Programming?
In the branch of mathematics concerned with the minimization or maximization of a linear function of several variables and inequalities; used in many branches of industry to minimize costs or maximize production.
How to use graphical method to solve Linear Programming?
To learn this let is refer to a problem of investment in tables and chairs discussed in Section 12.2. We will now solve this problem graphically. Let us graph the constraints stated as linear inequalities:
5x + y ≤ 100 ... (1)
x + y ≤ 60 ... (2)
x ≥ 0 ... (3)
y ≥ 0 ... (4)

The graph of this system (shaded region) consists of the points common to all half
planes determined by the inequalities (1) to (4) (Fig 12.1). Each point in this region
represents a feasible choice open to the dealer for investing in tables and chairs. The
region, therefore, is called the feasible region for the problem. Every point of this
region is called a feasible solution to the problem