A Geometric Approach to Linear Programming in the Two-Year College

Abstract

A(A) Typical introductory college mathematics courses taught in the two-year colleges to business and social science students often include aunit on linearprogramming. Although most students are able to master the mechanics of solving linear programming problems, they often have little understanding of the underlying theory. Valuable insights can be gained through the examination of a geometric model such as the one described in this paper. What is involved in the solution of a linear programming problem? Consider the following example: Two students, Bill and Allen, participate in a work-study program by working part time at a local art store doing sketches. Bill sketches two pictures per hour whereas Allen can sketch three pictures in the same length of time. Bill has a heavy course load and can't work more than ten hours per week. Allen needs the money and must work at least four hours per week. Because of equal opportunity regulations, Allen is not permitted to work more than one hour in excess of Bill and, because of a funding problem, the combined time the boys work cannot exceed thirteen hours. Suppose each week the store sells all but ten of the sketches. How many hours should the boys work to maximize the number of sketches sold, and how many sketches will be sold? To solve this type of problem a student ordinarily must set up and solve graphically a system of inequalities in two variables. He then defines a function on the solution set and, by examining the vertices of this set, discovers a solution that maximizes or minimizes the function. This solution may then be used to supply additional information. One difficulty encountered is setting up the inequalities. (The use of charts can be helpful when devised so that each row rep-