Linear, programming: Introduction (page 1 of 5 sections: Optimizing linear systems, Setting up word problems, linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. My first step is to solve each inequality for the more-easily graphed equivalent forms: It's easy to graph the system: Copyright Elizabeth Stapel All Rights Reserved To find the corner points - which aren't always clear from the graph - I'll pair the lines (thus forming a system of linear equations) and solve: y ( 1 / 2 ) x. So, to find the solution to this exercise, I only need to plug these three points into " z 3 x 4 y ". (2, 6 z 3(2) 4(6) (6, 4 z 3(6) 4(4) (1, 3 z 3(1) 4(3) Then the maximum of z 34 occurs at (6, 4), and the minimum of z 15 occurs at (1, 3). This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more. Advertisement, find the maximal and minimal value of z 3 x 4 y subject to the following constraints: The three inequalities in the curly braces are the constraints. The area of the plane that they mark off will be the feasibility region. The formula " z 3 x 4 y " is the optimization equation. I need to find the ( x, y ) corner points of the feasibility region that return the largest and smallest values of z.
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At (1, 3) So the corner points are (2, 6), (6, 4), and (1, 3). Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region.