Unit 1.6 – Linear Programming

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Unit 1.6 – Linear Programming
Ummmmm…yeah…I’m going to need you to go ahead and get out your notes…thanks..

Unit 1 – Algebra: Linear Systems, Matrices, & Vertex-Edge Graphs
1.6 – Linear Programming Georgia Performance Standard: MM3A6b – Represent and solve realistic problems using linear programming.

Vocabulary Linear Programming is a process of maximizing or minimizing a linear objective function. The objective function gives a quantity that is to be maximized (or minimized), and is subject to constraints. If all the constraints in a linear programming problem are graphed, the intersection of the graph is called the feasible region. If this region is bounded, then the objective function has a maximum value and a minimum value on the region. The maximum and minimum values each occur at a vertex of the feasible region.

What are some other words for constraints?
Restriction Limitation Control Limit Restraint Boundary How can we represent constraints mathematically? Inequalities <, >, ≤, and ≥

What’s the deal with linear programming
What’s the deal with linear programming? And when am I ever going to use this in real life? Business Piñatas Doughnuts Bikes Sunglasses Fast Cars Music If you like any of these (or anything in the entire world) you might use linear programming. Linear Programming lets us buy things we like and make the most of our money

Parts of Linear Programming
Objective Function What we are trying to minimize or maximize Ex. : C = 20x + 30y Constraints These are linear inequalities At least 3 Should intersect to form a shape called a feasible region Shade in!

Steps to Solve… Figure out what you’re minimizing or maximizing
This is your objective function List all your constraints Get the constraints into slope-intercept form Graph these Shade in the region Label the vertices These are the corners of the shapes Plug in the vertices to our objective function to find the best answer

Example Toy wagons are made to sell at a craft fair. It takes 4 hours to make a small wagon and 6 hours to make a large wagon. The owner of the craft booth will make a profit of \$12 for a small wagon and \$20 dollars for a large wagon. The craft booth owner has no more than 60 hours available to make wagons and wants to have at least 6 small wagons to sell. How many of each size should be made to maximize profit?

Example Find the minimum value and the maximum value of the objective function C = 3x + 2y subject to following constraints. x ≥ 0 y ≥ 0 x + 3y ≤ 15 4x + y ≤ 16

Guided Practice- Page 31: 2-4

Example Piñatas are made to sell at a craft fair. It takes 2 hours to make a mini piñata and 3 hours to make a regular sized piñata. The owner of the craft booth will make a profit of \$14 for each mini piñata sold and \$22 for each regular-sized piñata sold. If the craft booth owner has no more than 40 hours available to make piñatas and wants to have at least 16 piñatas to sell, how many of each size piñata should be made to maximize profit?

Homework