We can use graphical methods to solve linear optimization problems involving two variables. When there are two variables in the problem, we can refer to them as x1 and x2, and we can do most of the analysis on a two-dimensional graph. Although the graphical approach does not generalize to a large number of variables, the basic concepts of linear programming can all be demonstrated in the two-variable context. When we run into questions about more complicated problems, we can ask, what would this mean for the two-variable problem? Then, we can look for answers in the two-variable case, using graphs.
Another advantage of the graphical approach is its visual nature. Graphical methods provide us with a picture to go with the algebra of linear programming, and the picture can anchor our understanding of basic definitions and possibilities. For these reasons, the graphical approach provides useful background for working with linear programming concepts.
Consider the planning and scheduling problem facing a manufacturer of microwave ovens with two models in its line—the standard and the deluxe. Each oven is assembled from component parts and subassemblies that are produced in the mechanical and electronics departments. The following table shows the number of production hours per oven required in each department and the capacities of the three production departments, in monthly hours.