Presentation on theme: "Unit 1.6 – Linear Programming"— Presentation transcript:
1 Unit 1.6 – Linear Programming Ummmmm…yeah…I’m going to need you to go ahead and get out your notes…thanks..
2 Unit 1 – Algebra: Linear Systems, Matrices, & Vertex-Edge Graphs 1.6 – Linear ProgrammingGeorgia Performance Standard:MM3A6b – Represent and solve realistic problems using linear programming.
3 VocabularyLinear 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.
4 What are some other words for constraints? RestrictionLimitationControlLimitRestraintBoundaryHow can we represent constraints mathematically?Inequalities<, >, ≤, and ≥
5 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?BusinessPiñatasDoughnutsBikesSunglassesFast CarsMusicIf 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
6 Parts of Linear Programming Objective FunctionWhat we are trying to minimize or maximizeEx. : C = 20x + 30yConstraintsThese are linear inequalitiesAt least 3Should intersect to form a shape called a feasible regionShade in!
7 Steps to Solve… Figure out what you’re minimizing or maximizing This is your objective functionList all your constraintsGet the constraints into slope-intercept formGraph theseShade in the regionLabel the verticesThese are the corners of the shapesPlug in the vertices to our objective function to find the best answer
8 ExampleToy 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?
9 ExampleFind the minimum value and the maximum value of the objective function C = 3x + 2y subject to following constraints.x ≥ 0y ≥ 0x + 3y ≤ 154x + y ≤ 16
11 ExamplePiñatas 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?