The Simplex Method

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The Simplex Method
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Standard Linear Programming Problem

• Standard Maximization Problem
• 1. All variables are nonnegative.
• 2. All the constraints (the conditions) can be
  expressed as inequalities of the form
• ax by c, where c is a positive constant

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Illustrating Example (1)

• Maximize the objective function
• P(x,y) 5x 4y
• Subject to
• x y 20
• 2x y 35
• -3x y 12
• x 0
• y 0

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Solution
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What about when all of the constraints (the
inequalities) are of the type positive
constantBut we want to minimize the objective
function instead of maximizing.
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Minimization with constraintsIllustrating
Example (2)

• Minimize the objective function
• p(x,y) -2x - 3y
• Subject to
• 5x 4y 32
• x 2y 10
• x 0
• y 0

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Solution

• Let
• q(x) - p(x) - ( -2x -3y) 2x 3y
• To minimize p is to maximize q. Thus, we solve
  the following standard maximization linear
  programming problem
• Maximize the objective function
• q(x) 2x 3y
• Subject to
• 5x 4y 32
• x 2y 10
• x 0
• y 0

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• Rewriting the inequalities as equations, by
  introducing the slack variables u and v and the
  formula of the objective function as done in
  example (1).
• 5x 4y 32 , x 2y 10 and q 2x 3y
• Are transformed to
• 5x 4y u 32
• x 2y v 10
• - 2x - 3y q 0

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Standard Linear Programming Problem

• Standard Minimization Problem
• 1. All variables are nonnegative.
• 2. All the constraints (the conditions) can be
  expressed as inequalities of the form
• ax by c, where c is a positive constant

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• Solving
• The Standard Minimization Problem
• We use the fundamental theorem of Duality

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Illustrating Example (3)

• Minimize the objective function
• p(x,y) 6x 8y
• Subject to
• 40x 10y 2400
• 10x 15y 2100
• 5x 15y 1500
• x 0
• y 0

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• Minimize the objective function p(x,y) 6x
  8y
• Subject to
• 40x 10y 2400, 10x 15y 2100 , 5x
  15y 1500, x 0 and y 0
• We will refer to the above given problem by the
  primal (original) problem
• First We construct the following table, which we
  will refer to by the primal table
• x y constant
• ---------------------------------
• 40 10 2400
• 10 15 2100
• 5 15 1500
• ---------------------------------
• 6 8
• Second We construct a dual (twin) table from
  interchanging the rows and columns in the primal
  table
• x' y' z' constant
• --------------------------------------------------
  ---------
• 40 10 5 6
• 10 15 15 8

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• Fourth We apply the simplex method explained in
  example (1) to solve this problem
• Maximize the objective function q(x,y,z)
  2400x' 2100y' 1500z'
• Subject to
• 40x' 10y' 5z' 6, 10x' 15y' 15z' 8
  , x' 0 and y' 0, z' 0
• 4.a.Rewriting the inequalities and the formula of
  the objective function, with the slack variables
  being the same x and y (in that order) of the
  original (minimization) problem
• 40x' 10y' 5z' x 6
• 10x' 15y' 15z' y 8
• - 2400x' - 2100y' - 1500z q 0
• 4.b. We construct the simplex table for this
  problem

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Fifth We read the solution from the table
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Illustrating Example (4)

• Minimize the objective function
• p(x,y) x 2y
• Subject to
• -2x y 1
• - x y 2
• x 0
• y 0

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• Minimize the objective function p(x,y) x 2y
• Subject to
• -2x y 1, - x y 2 We will refer to the
  above given problem by the primal (original)
  problem
• First We construct the following table, which we
  will refer to by the primal table
• x y constant
• ---------------------------------
• -2 1 1
• -1 1 2
• ---------------------------------
• 1 2
• Second We construct a dual (twin) table from
  interchanging the rows and columns in the primal
  table
• x' y' constant
• -------------------------------------------
• -2 -1 1
• 1 1 2
• -----------------------------------------
• 1 2

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• Fourth We apply the simplex method explained in
  example (1) to solve this problem
• Maximize the objective function q( x ' , y )
  x' 2y'
• Subject to
• - 2x' - y' 1, x' y' 2 , x' 0 and y'
  0
• 4.a.Rewriting the inequalities and the formula of
  the objective function, with the slack variables
  being the same x and y (in that order) of the
  original (minimization) problem
• - 2x' - y' ' x 1
• x' y' y 2
• - x' - 2y' q 0
• 4.b. We construct the simplex table for this
  problem

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Homework

• 1. Using the simplex method, maximize p x
  (6/5)y subject to
• 2x y 180 , x 3y 300 , x 0 , y 0
• Solution p(48,84) 148.8
• 2. Minimize p(x,y) - 5x - 4y
• Subject to x y 20 , 2x y 35 , -3x y
  12 , x 0
• y 0
• Solution p(15,5) - 95
• 3. Using the dual theorem, minimize p 3x 2y
  subject to
• 8x y 80 , 8x 5y 240 , x 5y 100, x
  0 , y 0
• Solution p(20,16) 92
• Maximize the objective function