Nature does nothing uselessly. 3.4a: Linear Programming Nature does nothing uselessly.

What is Linear Programming Linear Programming is a technique that we use to identify a minimum or maximum value of some quantity. The objective function models that quantity. Constraints are the limits on the variables in our objective function. The feasible region of the graph contains all of the points that satisfy our constraints. Vertex Principle: If there is a minimum or maximum value of the linear objective function, it occurs at one or more vertices of the feasible region.

Vertex Principle Ex1) What values of x and y maximize P for the objective function P = 3x + 2y with constraints: 1) Graph your constraints. 2) What are the coordinates of each vertex? 3) Evaluate P for each vertex.

Vertex Principle Ex2) Find the values of x and y that maximize and minimize P if P = –5x + 4y. 1) Graph your constraints. 2) What are the coordinates of each vertex? 3) Evaluate P for each vertex.

Linear Programming ~ Application Ex3) A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make per day to maximize profit? How much is the maximum profit? 1) Define the variables. 2) Write the constraints and objective function.

Linear Programming ~ Application 3) Graph the constraints. 4) Find the coordinates of each vertex. P = 150x + 65y 5) Evaluate the objective function at each vertex.

Nature does nothing uselessly. 3.4a: Linear Programming Homework: 3.4 #1-9, p. 137# 40, 43, 50 Nature does nothing uselessly.