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

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Presentation on theme: "8/27: Linear Programming Lecture: LP Small Groups Homework."— Presentation transcript:

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. x 1, x 2, x 3,… : decision variables. So we write Max Z ( profit ) = (some combo of x 1...x X ) 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 x 1 = toy trucks, w/ a $7 profit each x 2 = dolls, w/ a $5 profit each x 3 = chess sets w/ a $12 profit each So Max Z (profit) = 7 x x x 3

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 x x x 3 <= 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 x x x 3 <= 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 x x x 3 <= 12000

17 Toy Company - constraints Other constraints: Integers:x 1, x 2, x 3 must be integers. Positive: x 1, x 2, x 3 >= 0

18 Toy Company - total LP Max Z (profit) = 7 x x x 3 S. T.: 10 x x x 3 <= x x x 3 <= x x x 3 <= x 1, x 2, x 3 >= 0 x 1, x 2, x 3 must be integers.

19 EX: Camping Trip. PCF $/lb beef jerky dried potatoes granola mix NutriGrain bars 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 x x 2 S. T. :4 x x 2 <= x x 2 <= 100 x 1. x 2 >= 0

23 LP - Graphical Solution 4 x x 2 = 240

24 LP - Graphical Solution 4 x x 2 = x x 2 = 100

25 LP - Graphical Solution 4 x x 2 = x x 2 = 100 Feasible Solution Region

26 LP - Graphical Solution 4 x x 2 = x x 2 = 100 Max Z = 7 x x 2 Z = $400 Z = $410 Z = $350


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