MATLAB is ideally suited to handle so-called linear programming problems. These are problems in which you have a quantity, depending linearly on several variables, that you want to maximize or minimize subject to several constraints that are expressed as linear inequalities in the same variables. If the number of variables and the number of constraints are small, then there are numerous mathematical techniques for solving a linear programming problem indeed, these techniques are often taught in high-school or university-level courses in finite mathematics. But sometimes these numbers are high, or, even if they are low, the constants in the linear inequalities or the object expression for the quantity to be optimized may be numerically complicated in which case a software package like MATLAB is required to effect a solution. We shall illustrate the method of linear programming by means of a simple example, giving a combined graphical/numerical solution, and then solve both a slightly as well as a substantially more complicated problem.

Suppose that a farmer has 75 acres on which to plant two crops: wheat and barley. To produce these crops, it costs the farmer (for seed, fertilizer, etc.) $120 per acre for the wheat and $210 per acre for the barley. The farmer has $15,000 available for expenses. But after the harvest, the farmer must store the crops while awaiting favorable market conditions. The farmer has storage space for 4000 bushels. Each acre yields an average of 110 bushels of wheat or 30 bushels...