# Graphical Linear Programming: Working with Constraints

## Understanding Constraints

• In Graphical Linear Programming, the term constraints refers to the limitations or conditions that can affect the possible solutions to an optimisation problem.
• Linear constraints are inequalities which involve a linear function. The solution to a linear constraint is a region of the coordinate plane that satisfies the inequality.

## Formulating the Constraints

• Constraints are typically formulated as inequalities. For example, in a practical problem concerning resource allocation, if there are limited resources, this might translate to a constraint such as x + y ≤ 100, where x and y represent units of different resources.
• When a constraint involves equality, it is referred to as an equation constraint. If it involves an inequality, it is an inequality constraint.
• Constraints can have a significant impact on solving a linear programming problem because they delineate the feasible region, where potential solutions can be found.

## Graphing the Constraints

• To plot a linear constraint, first treat the inequality as an equality. This gives a line on the coordinate plane.
• To determine which side of the line to shade (which represents the possible solutions), choose a test point not on the line. If the inequality is true for the chosen test point, shade that side of the line. Otherwise, shade the opposite side. This area is known as the feasible region.
• Where feasible regions overlapped by various constraints intersect, we find the overall feasible region and the potential solutions to the problem.

## Constraints in Practical Applications

• Linear programming is particularly useful in business and industry where constraints such as budgets, space, manpower, or materials often need to be optimised.
• In real-world problems, constraints often stipulate that certain variables must be non-negative, reflecting that factors like resources and time can’t be negative.

## Understanding the Feasible Region and Optimal Solution

• The feasible region, defined by the constraints, represents all possible solutions to the problem.
• The solution to the linear programming problem lies somewhere within this region. If the feasible region is unbounded, the linear programming problem may have no solution.
• The optimal solution to a problem is the point in the feasible region that optimises the objective function, which differs depending on whether the problem is a maximisation or minimisation problem.

## Key Points to Remember

• When using graphical methods to solve linear programming problems, the boundary of the feasible region corresponds to the equality part of the constraint.
• Constraints shape the feasible region, and it’s within this region that you’ll find the optimal solution.
• It’s crucial to correctly graph and interpret constraints, as without this understanding, you can’t identify the feasible region or optimal solution.
• Linear programming enables optimal solutions within specified constraints, lending its application to various fields where optimal resource utilisation is required.