Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 7 Systems of Equations and Inequalities 7.6 Linear Programming Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Write an objective function describing a quantity that must be maximized or minimized. Use inequalities to describe limitations in a situation. Use linear programming to solve problems.

Linear Programming Linear programming is a method for solving problems in which a particular quantity that must be maximized or minimized is limited by other factors. An objective function is an algebraic expression in two or more variables describing a quantity that must be maximized or minimized.

Example: Writing an Objective Function A company manufactures bookshelves and desks for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. The company’s profits are $25 per bookshelf and $55 per desk. Write the objective function that models the company’s total daily profit, z, from x bookshelves and y desks.

Constraints in Linear Programming In linear programming, the quantity that must be maximized or minimized is restricted by other factors, such restrictions are called constraints. Each constraint is expressed as a linear inequality. The list of constraints form a system of linear inequalities.

Example: Writing a Constraint A company manufactures bookshelves and desks for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. The company should not manufacture more than a total of 80 bookshelves and desks per day. Write an inequality that models this constraint.

Example: Writing a Constraint A company manufactures bookshelves and desks for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. To meet customer demand, the company must manufacture between 30 and 80 bookshelves per day, inclusive. Write an inequality that models this constraint.

Example: Writing a Constraint A company manufactures bookshelves and desks for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. To meet customer demand, the company must manufacture at least 10 and no more than 30 desks per day. Write an inequality that models this constraint.

Solving a Linear Programming Problem Let z = ax + by be an objective function that depends on x and y. Furthermore, z is subject to a number of constraints on x and y. If a maximum or minimum value of z exists, it can be determined as follows: 1. Graph the system of inequalities representing the constraints. 2. Find the value of the objective function at each corner, or vertex, of the graphed region. The maximum and minimum of the objective function occur at one or more of the corner points.

Example: Solving a Linear Programming Problem A company manufactures bookshelves and desks for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. The company’s profits are $25 per bookshelf and $55 per desk. The company should not manufacture more than a total of 80 bookshelves and desks per day. To meet customer demand, the company must manufacture between 30 and 80 bookshelves per day, inclusive. To meet customer demand, the company must manufacture at least 10 and no more than 30 desks per day. How many bookshelves and how many desks should be manufactured per day to obtain maximum profit? What is the maximum daily profit?

Example: Solving a Linear Programming Problem (continued) We must maximize subject to the following constraints: Step 1 Graph the system of inequalities representing the constraints.

Example: Solving a Linear Programming Problem (continued) Step 1 (cont) To maximize we must identify the vertices of the region that satisfies the system of constraints.

Example: Solving a Linear Programming Problem (continued) Step 1 (cont) To maximize we must identify the vertices of the region that satisfies the system of constraints. The vertices are solutions to the following systems of equations: solution (30, 30) solution (50, 30) solution (70, 10) solution (30, 10)

Example: Solving a Linear Programming Problem (continued) Step 2 Find the value of the objective function at each corner of the graphed region. The maximum or minimum of the objective function occur at one or more of the corner points. Corner (x, y) Objective function (30, 30) z = 25(30) + 55(30) = 2400 (30, 10) z = 25(30) + 55(10) = 1300 (50, 30) z = 25(50) + 55(30) = 2900 (70, 10) z = 25(70) + 55(10) = 2300

Example: Solving a Linear Programming Problem (continued) A company manufactures bookshelves and desks for computers. The company’s profits are $25 per bookshelf and $55 per desk. The constraints are that the company should not manufacture more than a total of 80 bookshelves and desks per day; the company must manufacture between 30 and 80 bookshelves per day, inclusive; and the company must manufacture at least 10 and no more than 30 desks per day. How many of each should be manufactured per day to obtain maximum profit? What is the maximum daily profit? 50 bookshelves and 30 desks must be manufactured per day to obtain the maximum profit of $2900.