Algebra 2 Notes Systems of Equations and Inequalities Unit 03b. Optimization with Linear Programming

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1 Optimization with Linear Programming Big Idea Linear programming is one of the most practical uses of mathematics in the real world. The inequalities of the system represent the constraints in the problem and once graphed define the feasible region. The intersection points of the lines that define the polynomial region are known as the vertices. The coordinates of the vertices can be used to maximize or minimized a specific objective function. The process of maximizing or minimizing the objective function is known as optimization. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Find the maximum and minimum values of a function over a region Solve real-world optimization problems using linear programming Vocabulary Linear Programming - The process of finding the maximum or minimum values of a function for a region defined by the inequalities. Feasible Region - The intersection of the graphs in a system of constraints. Bounded - A region is bounded when the graph of a system of constraints is polynomial region. Unbounded - A system of inequalities that forms a region that is open. Optimize- To seek the optimal price or amount that is desired to minimize cost or maximize profit. Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 1 of 7 6/20/2013

2 Examples, Notes, and Exam Questions Goal: Learn to solve linear programming problems and apply it to real-life situations. Linear programming is the process of optimizing a linear objective function subject to a system of linear inequalities called the constraints. The graph of the system can be bounded or unbounded and is referred to as the feasible region. Optimization is a process of finding the maximum or minimum value of some quantity. Objective Function is the known expression to be optimized. Constraints are the linear inequalities which are graphed to form the feasible region and whose intersection points (vertices) define the maximum are minimum values. The Feasible Region is the shaded area of the system which contains all possible solutions to the system of constraints. A feasible region that is enclosed on all sides by constraints is known as a bounded region. A feasible region with at least one side not constrained is known as an unbounded region. The maximum and minimum values of the objective function are found by substituting the ordered pairs of the vertex points into the objective function. Steps for setting up and solving a linear programming problem 1 st. Graph the linear inequalities (constraints) on one plane. 2 nd. Indentify the feasible region (bounded or unbounded) and the vertices (the intersection points of the inequalities). 3 rd. Evaluate the objective function by substituting the x and y coordinates of the vertices into the objective function. 4 th. Determine the maximum and minimum values of the function. Note: Most linear programming problems will have feasible regions constrained in the first quadrant of the plane. Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 2 of 7 6/20/2013

3 EX 1: Find the minimum and maximum values of the objective function subject to the given constraints. Objective Function: C 2x y x 5 x 0 Constraints: y 2 y 2 EX 2: Find the minimum and maximum values of the objective function subject to the given constraints. Objective Function: C 2x y x 0 y 0 Constraints: x y 7 5x 2y 20 Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 3 of 7 6/20/2013

4 Real life problem EX 3: An office manager is purchasing file cabinets and wants to maximize storage space. The office has 60 square feet of floor space for the cabinets and $600 in the budget to purchase them. Cabinet A requires 3 square feet of floor space, has a storage capacity of 12 cubic feet, and costs $75. Cabinet B requires 6 square feet of floor space, has a storage capacity of 18 cubic feet, and costs $50. How many of each cabinet should the office manager buy? Objective Function: Constraints: Use the feasible region to answer the following questions. a. What are the vertices of the feasible region? b. What are the minimum and maximum of the objective function? C 2x 3y Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 4 of 7 6/20/2013

5 SAMPLE QUESTIONS 1. Find the minimum and maximum values of the objective function subject to the given constraints. Objective function: C x 4y Constraints: x 2 x 5 y 1 y 6 2. Find the minimum and maximum values of the objective function for the given feasible region. Objective Function: C 2x y 3. Find the minimum and maximum values of the objective function subject to the given constraints. Objective function: C 2x y Constraints: x 0 y x 2 y x 6 Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 5 of 7 6/20/2013

6 4. The drama club is having a car wash as a fundraiser. They wash x cars at $5 each and y trucks at $8 each. They will wash at least 26 cars and 15 trucks. They need to make at least $250. Which system describes this situation? D 5. A factory makes widgets x and gadgets y. Widgets cost $3 and gadgets cost $5 each to make. The boss wants at least 10 widgets and 20 gadgets per day. Total costs need to be at most $300 per day. What system represents this situation? C 6. A class fundraiser is earning $7 per car wash w and $4 per cake c sold. The cakes cost $2 to make and the supplies or the car wash cost $1 per car. They will wash less than 50 cars and bake no more than 30 cakes. They want to raise at least $320 to give to a local charity. What system represents this situation? D Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 6 of 7 6/20/2013

7 7. Bianca is buying s shirts and p pairs of pants for the new school year. Shirts cost $10 each and pants cost $15 each. She needs at least 4 shirts and 2 pairs of pants, and can spend no more than $100. Which system represents this situation? A 8. Maximize the objective function under the constraints. A At Healthy Hair, the cost of a children s haircut x is $4 and the cost of an adult haircut y is $14. The manager s goal for the day is to have at least 5 children s cuts, at least 20 adult cuts, and to have total sales be greater than $500. Write a system to represent the situation. Algebra 2 Unit 03b Optimization - Systems of Equations and Inequalities Page 7 of 7 6/20/2013