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1. The Problem 2. Tabulate Data 3. Translate the Constraints 4. The Objective Function 5. Linear Programming Problem 6. Production Schedule 7. No Waste 8. Feasible Set 1

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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? 2

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It is helpful to tabulate data given in the problem. ChairSofaAvailable time Carpentry Finishing Upholstery Profit 6 hours3 hours96 labor-hours 1 hour 18 labor-hours 2 hours6 hours72 labor-hours $80$70 3

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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. 4

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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) = 6 x + 3 y [number of labor-hours per day] < [maximum available] 6 x + 3 y < 96 5

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Similarly, Finishing: x + y < 18 Upholstery: 2 x + 6 y < 72 Number of chairs and sofas cannot be negative: x > 0, y > 0 6

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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] = 80 x + 70 y 7

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

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

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Which of the following is a production schedule for (11,6)?(6,11)? 10 YesNo

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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 11

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

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

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Find the feasible set for 14

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

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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. 16

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