Linear Programming

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Linear Programming
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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 worlds
  greatest cities.

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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.

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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.

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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.

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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.

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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

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Constraints

• x bottled water y medical kits
• x 0
• y 0
• 20x 10y 0
• 1x 1y 0

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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.