Graphical Linear Programming: Graphical Solutions

Introduction

In linear programming, graphical solutions offer a practical visual representation to solve systems of linear inequalities.
These methods are typically used when there are two variables, as they can be easily represented graphically.

A graphical solution to a linear programming problem involves sketching the feasible region, which is the area that satisfies all constraints.
Each constraint is represented as a line on the graph, and their intersection forms the feasible region.
The objective function, which is the function to be maximised or minimised, is also plotted on the same graph.
The solution to the problem is the point(s) in the feasible region that gives the maximum or minimum value of the objective function.

To plot the graph, start by drawing the lines corresponding to each constraint. This is done by treating each inequality as an equality to find the line, then shading the area that satisfies the inequality.
Where the shaded regions overlap forms the feasible region.
The feasible region is either a polygon (bounded) or extends infinitely in one or more directions (unbounded).
Note that some problems may have no feasible region, meaning that there is no solution that satisfies all constraints.

Once the feasible region is plotted, draw a line that represents the objective function.
Move this line parallel to itself within the feasible region in the direction that maximises or minimises the function as required.
The point where this line is farthest from the origin while still touching the feasible region is the optimal solution.
Remember that in linear programming, the optimal solution often lies at a vertex or corner point of the feasible region.

The coordinates of the optimal solution represent the values of the variables that maximise or minimise the objective function.
The value of the objective function at this solution point is the maximum or minimum possible value that can be achieved.
When the feasible region is unbounded and the objective function is to be maximised, there may not be a single optimal solution. The function might increase without limit.

A typical example of graphical linear programming involves optimising the profit or cost in manufacture, given constraints on available resources.
Another common application is in diet planning, where the goal might be to minimise cost while following guidelines on nutritional requirements.