Linear Programming

1. Linear Programming

2. Linear Programming • The Berlin Airlift (1948-1949) was an operation by the United States and Great Britain in response to military action by the former Soviet Union: Soviet Troops closed all roads and rail lines between West Germany and Berlin cutting off supply routes to the city. The Allies used mathematical technique developed during World War II to maximize the amount of supplies transported. During the 15-month airlift, 278,228 flights provided basic necessities to blockade Berlin, saving one of the world’s greatest cities.

3. Linear Programming • We are going to look at an important application of systems of linear inequalities. Such systems arise in linear programming, a method for solving problems in which a particular quantity that must be maximized or minimized is limited by other factors. Linear programming is one of the most widely used tools in management science. It helps businesses allocate resources to manufacture products in a way that will maximize the profit. Linear programming accounts for more that 50% and perhaps as much as 90% of all computing time used for management decisions in business. The Allies used linear programming to save Berlin.

4. Linear Programming Objective Function • Many problems involve quantities that must be maximized or minimized. Businesses are interested in maximizing profit. An operation in which bottled water and medical kits are shipped to earthquake victims needs to maximize the number of victims helped by this shipment. An objective function is an algebraic expression in two or more variables describing a quantity that must be maximized or minimized.

5. Objective Function • Bottled water and medical supplies are to be shipped to victims of an earth-quake by plane. Each container of bottled water will serve 10 people and each medical kit will aid 6 people. If x represents the number of bottles of water to be shipped and y represents the number of medical kits, write the objective function that describes the number of people that can be helped.

6. Objective Function • Solution Because each bottle of water serves 10 people and each medical kit aids 6 people, we have • The number of people helped is • = 10x + 6y. • Using z to represent the objective function, we have • z = 10x + 6y. • Unlike the functions that we have seen so far, the objective function is an equation in three variables. For a value of x and a value of y, there is one and only one value of z. Thus, z is a function of x and y.

7. Constraints • Each plane can carry no more than 80,000 pounds. The bottled water weighs 20 pounds per container and each medical kit weighs 10 pounds. • Each plane can carry a total volume of supplies that does not exceed 6000 cubic feet. Each water bottle is 1 cubic foot and each medical kit also has a volume of 1 cubic foot. • Write inequalities to represent both of these constraints

8. Constraints • x = bottled water y = medical kits • x ≥ 0 • y ≥ 0 • 20x + 10y ≤ 0 • 1x + 1y ≤ 0

9. Solving a Linear Programming Problem • Let z = ax + by be an objective function that depends of x and y. Z is subject to a number of constraints on x and y. If a maximum or minimum value of z exists, it can be determined as follows: • 1. Graph the system of inequalities representing the constraints • 2. Find the value of the objective function at each corner, or vertex, of the graphed region. The maximum and minimum of the objective function occur at one or more of the corner points.