Chapter 5 The Simplex Method

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Chapter 5The Simplex Method
The most popular method for solving Linear
Programming Problems We shall present it as an
Algorithm
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General Structure of Algorithms
Initialise
Check for desired results
Yes
Stop
Iterate
No
Perform a sequence of repetitive steps
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Construct a feasible extreme point
Is this point optimal ?
Yes
Stop
Iterate
No
Move along an edge to a better extreme point
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Missing Details

• Initialisation
• How do we represent a feasible extreme point
  algebraically?
• Optimality Test
• How do we determine whether a given extreme point
  is optimal?
• Iteration
• How do we move a long an edge to a better
  adjacent extreme point?

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5.1 initialisation

• Transform the LP problem given in a standard form
  into a canonical form.
• This involves the introduction of slack
  variables, one for each functional constraint.
• Thus if we start with n variables and m
  functional constraints, we end up with nm
  variables and m functional equality constraints.

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Standard Form

• optmax
• ???
• bi 0 , for all i.

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Canonical Form
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Observation

• The i-th slack variable measure the distance of
  the point x(x1,...,xn) from the hyperplane
  defining the i-th constraint (This is not a
  Euclidean distance).
• Thus, if the i-th slack variable is equal to zero
  the point x (x1,...,xn) is on the i-th
  hyperplane. Otherwise it is not.
• The original variables measure the distance to
  the hyperplanes defining the respective
  non-negativity constraints.

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Example
x3,x4,x5 are slack variables
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Why do we do this?

• If we use the slack variables as a basis, we
  obtain a feasible extreme point !!!

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5.5.1 Definition

• A basic feasible solution is a basic solution
  that satisfies the non-negativity constraint.
• Observation
• A basic feasible solution is an extreme point of
  the feasible region.
• Thus
• Initialisation involves constructing a basic
  feasible solution using the slack varaibles.

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Example
x3,x4,x5 are slack variables

• Initial basic feasible solution x
  (0,0,40,30,15), namely
• x1 0 x2 0 x3 40 x4 30 x5 15

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Summary of the Initialisation Step

• Select the slack variables as basic
• Comments
• Simple
• Not necessarily good selection the first basic
  feasible solution can be (very) far from the
  optimal solution.