Write your objective function. Make sure that you write the function in terms of objective variables, such as time, products or labor. For example, if you run a print shop, you may have an objective function that looks like "Z = 15b + 10p" where "b" stands for the units of business cards sold and "p" stands for the units of pamphlets sold in a day. This function shows the point where your profit will be maximized.
Write the constraints. The constraints are always in terms of the variables in the objective function and should be written as inequalities. For example, your print shop may only have one machine that can print business cards and pamphlets, but you also have an implicit time constraint. This means you cannot use the machine whenever you want. It may be the case that your constraint functions are "b < 2," "p <3" and "b + p < 4." Whether you use "less than" signs or "less than or equal to" signs does not matter, as you will be working with continuous numbers for linear programs, yielding the same results.
Find the feasible region. This is the area in which all the constraints are satisfied. You can sketch this region if needed. If you wish to sketch the example problem, sketch "b < 2," "p <3" and "b + p < 4." The resulting region is the feasible region.
Find the vertices of the feasible region. This is where the lines of the constraint functions meet. For the example, the vertices are the points (2, 2) and (1, 3).
Determine the maximum profit. Plug the values of the vertices into the objective function and check the resulting values. The resulting value that is the largest in the maximum profit. In the example, you get 50 for the point (2, 2) and 45 for the point (1, 3). Thus, 50 is the maximum profit.