Linear Programming

Linear Programming

Linear Programming (LP) is a mathematical technique used to find the best possible solution to a problem involving limited resources. It helps determine the optimal value (maximum profit, minimum cost, maximum production, etc.) while satisfying a set of constraints.

In simple terms, linear programming answers questions such as:

• How can a factory maximize profit using limited raw materials?
• How can a company minimize transportation costs?
• How should resources be allocated to achieve the best outcome?

Inequality and graph

The mathematical statement with any of these symbols \(\lt, \gt, \leq, \geq \neq\) is called an inequality. The inequality of degree 1 is called linear inequality.

Examples:

• One variable: \(x \geq 0\), \(y \leq 4\)
• Two variable: \(x + y \leq 0\), \(2x – y \geq 4\)

System of Linear Inequalities

When two or more linear inequalities are represented in the same graph, the common solution region in the shape of a convex polygon is called a system of linear inequalities. The vertices of the common solution region of this system of linear inequalities satisfy all the given linear inequalities. The common solution region is also called feasible Region.

Solution of Linear Programming Problems

Linear Programming (LP) is a mathematical technique used in business, industry, and trade to make the best use of resources, reduce costs, and increase profits. It helps determine the maximum or minimum value of a linear function while satisfying certain conditions.

• The function to be optimized (maximized or minimized) is called the objective function.
• The conditions or restrictions are called constraints, which are expressed as inequalities.

There are two methods for solving linear programming problems:

• Graphical Method
• Simplex Method

At this level, only the graphical method is used.

Steps in the Graphical Method

1. Plot all given linear inequalities on the same graph.
2. Identify the common solution region, called the feasible region.
3. Determine the coordinates of the vertices (corner points) of the feasible region.
4. Substitute these coordinates into the objective function.
5. Choose the maximum value if the objective is maximization, or the minimum value if the objective is minimization.