A2 ­ 3.4 day 1 ­ intro to linear programming.notebook
December 05, 2013
Algebra 2
3.4 ‐ Intro to Linear Programming A. Intro: The Berlin Airlift of 1948 was an operation put forth by the United States and Great Britain in response to military action by the former Soviet Union. The Soviet Union had cut off all roads and rail lines between West Germany and Berlin, creating a major problem for the people of Berlin. The Allies used a mathematical technique known as Linear Programming that was developed during World War II to maximize the amount of relief supplies transported to the people of Berlin. During the 15‐month blockade, 278,228 flights from Allied troops to Berlin delivered the much needed, basic necessity and supplies to the country. The Berlin Airlift saved one of the world’s greatest cities; and Linear Programming helped to make it possible. Nov 19­9:04 AM
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A2 ­ 3.4 day 1 ­ intro to linear programming.notebook
December 05, 2013
B. Definition
• Linear Programming ‐ a method for solving problems in which a particular quantity that must be maximized or minimized is limited by other factors. • In the example above, the Allies wanted to maximize basic supplies of food, water, clothing and medical supplies but would have been limited by factors such as the available space on the plane and the weight of carry‐on items that the plane could withstand. Nov 19­9:04 AM
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A2 ­ 3.4 day 1 ­ intro to linear programming.notebook
December 05, 2013
C. The Objective Function
• The Objective Function is the equation in two or more (mostly 3) variables that describes the quantity that which is trying to be maximized or minimized. • In order to be successful at Linear Programming, students must be able to identify and write the objective function describing a quantity that must be maximized or minimized. Nov 19­9:04 AM
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A2 ­ 3.4 day 1 ­ intro to linear programming.notebook
December 05, 2013
D. Examples: Writing the Objective Function
1. Bottled water and medical supplies are to be shipped to victims of an earthquake by plane. Each bottle of water will serve 10 people, while each medical kit will aid 6 people. Write the objective function that describes the number of people that can be helped. 2. A company manufactures bookshelves and desks for computers. The company’s profits are $25.00 per bookshelf and $55.00 per desk. Write the objective function that describes the profits of bookshelves and desks. Nov 19­9:05 AM
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A2 ­ 3.4 day 1 ­ intro to linear programming.notebook
December 05, 2013
E. Writing Constraints ‐ the limiting factors
Ideally, the number of earthquake victims helped in example 1 should •
increase without any restrictions so that everyone would receive water and medical attention. However, due to limiting factors such as weight restrictions of the plane and volume (or space) restrictions of the plane, an infinite amount of supplies cannot be shipped. Thus, an infinite amount of people cannot be helped. In linear programming, these limiting factors are called constraints and are written as inequalities. Together, along with the objective function, a system of linear equations and inequalities has been formed. Nov 19­9:05 AM
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A2 ­ 3.4 day 1 ­ intro to linear programming.notebook
December 05, 2013
F. Examples. Writing the constraints.
(refer to example 1)Each plane can carry no more than 80,000 lbs. The bottled water weighs 20 pounds per container and each medical kit weighs 10 pounds. Furthermore, each plane can carry a volume of supplies that does not exceed 6,000 cubic feet. Each bottle takes up 1 cubic foot of space and each medical kit also requires 1 cubic foot of volume. Write two inequalities that model the constraints placed upon weight and volume. (refer to example 2)To meet customer demand, the company must manufacture between 30 and 80 bookshelves per day inclusive. Furthermore, the company must manufacture at least 10 and no more than 30 desks per day. Write the inequalities that model the constraints on the company based upon consumer demand. Nov 19­9:05 AM
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