Linear Programming

Linear Programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model. Its function is to maximize or minimize a linear objective function, subject to linear equality and inequality constraints.

The objective function is a mathematical expression that defines the goal of the linear programming problem, typically to maximize or minimize a quantity, such as profit or cost.

The feasible region is the set of all possible points that satisfy the constraints of a linear programming problem. It is typically represented graphically as a polygon.

Corner points (or vertices) of the feasible region are significant because the optimal solution to a linear programming problem will occur at one of these points.

A constraint is a condition that must be satisfied in a linear programming problem, usually expressed as a linear inequality or equation.

Dashed lines are used to represent inequalities that do not include the boundary (e.g., < or >), while solid lines are used for inequalities that do include the boundary (e.g., ≤ or ≥).

To determine the maximum value of an objective function, evaluate the function at each corner point of the feasible region and identify the highest value.

A solid line indicates that the points on the line are included in the solution set (≥ or ≤), while a dashed line indicates that the points on the line are not included (> or <).

A point outside the feasible region does not satisfy all the constraints of the linear programming problem and is therefore not a viable solution.

The graphical method is a technique used to solve linear programming problems with two variables by graphing the constraints, identifying the feasible region, and evaluating the objective function at the corner points.