Graphical Method of Solving Linear Programming Definition: The graphical method is a graphical approach to solving linear programming problems. It invol...
Graphical Method of Solving Linear Programming
Definition:
The graphical method is a graphical approach to solving linear programming problems. It involves visualizing the feasible region (graph of the linear inequalities), identifying the vertices of the region, and determining the optimal solution by finding the coordinates of the vertices that minimize or maximize the objective function.
Steps:
Create a scatter plot of the decision variables (x and y).
Define the linear inequalities as constraints, represented by lines or rays.
The feasible region is the area where the points satisfying the inequalities lie.
The vertices of the feasible region are the points where the lines representing the constraints intersect.
These points correspond to the corner points of the feasible region.
The optimal solution is the vertex that minimizes or maximizes the objective function.
The objective function is typically represented by a single variable (e.g., z = x + y) or a combination of two variables (e.g., z = 2x - y).
The coordinates of the optimal solution provide the values of x and y that maximize or minimize the objective function.
The value of the objective function at the optimal solution is the maximum or minimum value of the function.
The feasible region represents the set of all possible solution points, and the vertices represent the extreme points of this set.
Example:
Suppose we have the following linear programming problem:
Maximize z = 3x + 4y
Subject to:
x + y ≤ 5
2x - y ≥ 0
x ≥ 0
The corresponding linear inequalities are:
x + y ≤ 5
2x - y ≥ 0
x ≥ 0
The feasible region is a triangle in the first quadrant (x, y) such that x + y is less than or equal to 5 and 2x - y is greater than or equal to 0.
The vertices of the feasible region are (0, 5), (2, 3), and (5, 0). The optimal solution is at (2, 3), with the objective function value of 15.
The feasible region is represented by the lines x + y = 5 and 2x - y = 0, and the vertices are marked on a coordinate plane