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8/27: Linear Programming

8/27: Linear Programming. Lecture: LP Small Groups Homework. Linear Programming. What is it? Synthesizing a problem in words into a series of equations. A type of modeling tool Optimizing a linear function subject to several constraints, expressed as inequalities. LP - 4 Characteristics.

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8/27: Linear Programming

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  1. 8/27: Linear Programming • Lecture: LP • Small Groups • Homework

  2. Linear Programming • What is it? • Synthesizing a problem in words into a series of equations. • A type of modeling tool • Optimizing a linear function subject to several constraints, expressed as inequalities.

  3. LP - 4 Characteristics • Objective Function • Constraints • Alternative Courses of Action • Linear Equations

  4. EX: Toy Company • A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively.

  5. Toy Company Formulate a linear program set to maximize the company's profit.

  6. Terminology • Z : variable to be optimized. • x1, x2, x3,… : decision variables. So we write Max Z ( profit ) = (some combo of x1...xX) S. T. ("subject to"): (the constraints)

  7. Toy Company • What are we supposed to maximize? • What factors play a part in that? • What constraints are there to the profit?

  8. A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. • Maximize the company’s profit.

  9. A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. • Maximize the company’s profit.

  10. A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. • Maximize the company’s profit.

  11. A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hrs. of hand labor time, 8 hrs. machine time, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. • Maximize the company’s profit.

  12. Toy Company • What are we supposed to maximize? • THE PROFIT • What factors play a part in that? • PROFIT FROM TRUCKS, DOLLS, and CHESS SETS • What constraints are there to the profit? • HAND TIME, MACHINE TIME, and WOOD

  13. Toy Company • Let x1 = toy trucks, w/ a $7 profit each • x2 = dolls, w/ a $5 profit each • x3 = chess sets w/ a $12 profit each • So Max Z (profit) = 7 x1 + 5 x2 + 12 x3

  14. Toy Company - constraints • Hand Time: total of 8 hours. -- or 480 min. • Truck - 10 min. • Doll - 8 min. • Chess Set - 3 min. • so 10 x1 + 8 x2 + 3 x3 <= 480

  15. Toy Company - constraints • Machine Time: total of 8 hrs. -- or 480 min. • Truck - 3 min. • Doll - 10 min. • Chess Set - 20 min. • so 3 x1 + 10 x2 + 20 x3 <= 480

  16. Toy Company - constraints • Wood: total of 1000 ft. -- or 12,000 in. • Truck - 15 in. • Doll - 11 in. • Chess Set - 31 in. • so 15 x1 + 11 x2 + 31 x3 <= 12000

  17. Toy Company - constraints • Other constraints: • Integers: x1, x2, x3 must be integers. • Positive: x1, x2, x3 >= 0

  18. Toy Company - total LP • Max Z (profit) = 7 x1 + 5 x2 + 12 x3 S. T.: 10 x1 + 8 x2 + 3 x3 <= 480 3 x1 + 10 x2 + 20 x3 <= 480 15 x1 + 11 x2 + 31 x3 <= 12000 x1, x2, x3 >= 0 x1, x2, x3 must be integers.

  19. EX: Camping Trip. P C F $/lb beef jerky 10 4 8 13.00 dried potatoes 0 12 2 2.50 granola mix 4 8 11 8.50 NutriGrain bars 2 14 5 9.00 Must have 30 g. protein, 60 g. carbohydrates, and 15 g. of fat. Minimize the cost.

  20. Graphical Solutions for LP • Sparky Electronics • 2 products, WalkFM & WristTV • profit: $7 $5 • machine time 4 3 • assembly time 2 1 • Total machine time 240 • Total assembly time 100

  21. LP - Graphical Solution • Limitation to the method: only TWO decision variables can exist.

  22. LP - Graphical Solution Maximize Z ( profit ) = 7 x1 + 5 x2 S. T. : 4 x1 + 3 x2 <= 240 2 x1 + 1 x2 <= 100 x1 .x2 >= 0

  23. LP - Graphical Solution 4 x1 + 3 x2 = 240

  24. LP - Graphical Solution 4 x1 + 3 x2 = 240 2 x1 + 1 x2 = 100

  25. LP - Graphical Solution 4 x1 + 3 x2 = 240 2 x1 + 1 x2 = 100 Feasible Solution Region

  26. LP - Graphical Solution Z = $400 4 x1 + 3 x2 = 240 2 x1 + 1 x2 = 100 Max Z = 7 x1 + 5 x2 Z = $410 Z = $350

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