Linear Programming Intro Notes
Some students make necklaces and bracelets in their spare time and sell all that they make. Every week they have available 10,000 g of metal and 20 hours to work. It takes 50 g of metal to make a necklace and 200 g to make a bracelet. Each necklace takes 30 minutes to make and each bracelet takes 20 minutes. The profit on each necklace is $3.50 and each bracelet is $2.50. The students want to earn as much money as possible. Because you are taking this algebra course, they ask you to give them advice. What numbers of necklaces and bracelets should they make each week?
we'll solve this in a few slides...
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Linear Programming Intro Notes
In 1826, the French mathematician Joseph Fourier proved the following theorem:
Linear ­ Programming Theorem:
The feasible region of a linear programming problem is convex, and the maximum or minimum quantity is determined at one of the vertices of the region.
The Linear­Programming Theorem tells you where to look for the greatest or least value of a linear combination expression in a linear­
programming situation, without having to draw many lines through the feasible region. (More on this later.)
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Linear Programming Intro Notes
World War II and Linear Programming
Need for efficient transportation of supplies
George Dantzig develops what is called the simplex method.
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Linear Programming Intro Notes
"for their contributions to the theory of optimum allocation of resources"
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Linear Programming Intro Notes
Linear programming is often used in industries in which all the competitors make the same product (such as oil, paper, milk, and so on). Efficiency in the use of labor and materials determines the amount of profit. These situations can involve as many as 5000 variables and 10,000 inequalities. Computers would be used to solve such a complex system.
1984: Naredra Karmarkar
developed a time saving linear programming method at Bell Laboratories in New Jersey
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Linear Programming Intro Notes
The following procedure can be used to solve linear programming problems.
1.
2.
3.
4.
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6.
Define variables. Write the constraints as a system of inequalities.
(restrictions)
Graph the system and find the coordinates of the vertices of the polygon formed. (feasible region)
Write an expression to be maximized or minimized. (objective function)
Substitute values from the coordinates of the vertices into the expression.
Select the greatest or least result. (optimum point)
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Linear Programming Intro Notes
Some students make necklaces and bracelets in their spare time and sell all that they make. Every week they have available 10,000 g of metal and 20 hours to work. It takes 50 g of metal to make a necklace and 200 g to make a bracelet. Each necklace takes 30 minutes to make and each bracelet takes 20 minutes. The profit on each necklace is $3.50 and each bracelet is $2.50. The students want to earn as much money as possible. Because you are taking this algebra course, they ask you to give them advice. What numbers of necklaces and bracelets should they make each week?
x = # of necklaces
y = # of bracelets
grams
minutes
necklace
50
30
bracelet
200
20
constraints
1. 50x + 200y ≤ 10000
2. 30x + 20y ≤ 1200
3. x ≥ 0, y ≥ 0
graphing from standard form gives you an idea of the scale needed on the x and y axis
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Linear Programming Intro Notes
Jewelry Making
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Linear Programming Intro Notes
Profit Equation (objective function):
3.5x + 2.5 y = P
(x, y)
3.5x + 2.5y = P
(0, 0)
(0, 50)
(40, 0)
(8, 48)
3.5(0) + 2.5(0) = 0
3.5(0) + 2.5(50) = 125
3.5(40) + 2.5(0) = 140
3.5(8) + 2.5(48) = 148
8 necklaces and 48 bracelets maximizes profits
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