3.1 A Linear Programming Problem

1. 3.1 A Linear Programming Problem • The Problem • Tabulate Data • Translate the Constraints • The Objective Function • Linear Programming Problem • Production Schedule • No Waste • Feasible Set

2. The Problem • A furniture manufacturer makes two types of furniture - chairs and sofas. The manufacture of a chair requires 6 hours of carpentry, 1 hour of finishing, and 2 hours of upholstery. Manufacture of a sofa requires 3 hours of carpentry, 1 hour of finishing, and 6 hours of upholstery. Each day the factory has available 96 labor hours for carpentry, 18 labor-hours for finishing, and 72 labor-hours for upholstery. The profit per chair is $80 and per sofa is $70. How many chairs and sofas should be produced each day to maximize the profit?

3. Tabulate Data • It is helpful to tabulate data given in the problem.

4. Translate the Constraints • Translate each of the constraints (restrictions on labor-hours available) into mathematical language. • Let x be the number of chairs and y be the number of sofas manufactured each day, respectively.

5. Translate the Constraints (2) • Carpentry: [number of labor-hours per day] • = (number of hours required per chair)  (number of chairs per day) + (number of hours required per sofa)  (number of sofas per day) • = 6x + 3y • [number of labor-hours per day] < [maximum available] • 6x + 3y< 96

6. Translate the Constraints (3) • Similarly, • Finishing: • x + y<18 • Upholstery: • 2x + 6y< 72 • Number of chairs and sofas cannot be negative: • x> 0, y> 0

7. The Objective Function • The objective of the problem is to optimize profit. Translate the profit (objective function) into mathematical language. • [profit] = [profit from chairs] + [profit from sofas] • = [profit per chair][number of chairs] + [profit per sofa][number of sofas] • = 80x + 70y

8. Linear Programming Problem • The manufacturing problem can now be written as a mathematical problem. • Find x and y for which 80x + 70y is as large as possible, and for which the following hold simultaneously: This is called a linear programming problem.

9. Production Schedule • In the manufacturing problem, each pair of numbers (x,y) that satisfies the system of inequalities is called a production schedule.

10. Example Production Schedule • Which of the following is a production schedule for • (11,6)? (6,11)? Yes No

11. No Waste • It seems clear that a factory will operate most efficiently when its labor is fully utilized (no waste). • This would require x and y to satisfy the system

12. Example No Waste • Solve According to the graph of the three equations, there is no common intersection and therefore no solution.

13. Feasible Set • The set of solutions to the system of inequalities is called the feasible set of the system. This represents all possible production schedules.

14. Example Feasible Set • Find the feasible set for

15. Example Feasible Set (2) • Notice that (0,0) satisfies all the inequalities. • Graph the boundaries: • y< -2x + 32 • y< -x + 18 • y< -x/3 + 12 • x> 0, y> 0 Feasible Set

16. Summary Section 3.1 • A linear programming problem asks us to find the point (or points) in the feasible set of a system of linear inequalities at which the value of a linear expression involving the variables, called the objective function, is either maximized or minimized.