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# L16 - CO350 Linear Programming Chapter 6 The Simplex Method...

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CO350 Linear Programming Chapter 6: The Simplex Method 8th June 2005

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Chapter 6: The Simplex Method 1 Minimization Problem ( § 6.5) We can solve minimization problems by transforming it into a maximization problem. Another way is to change the selection rule for entering variable. Since we want to minimize z , we would now choose a reduced cost ¯ c k that is negative , so that increasing the nonbasic variable x k decreases the objective value. We now stop with an optimal solution only when ¯ c j 0 for all j N . Final remark: We can always transform the LP into a maximization problem, and use the simplex method as normal.
Chapter 6: The Simplex Method 2 Choice Rules ( § 6.6) In the simplex method, we need to make two choices at each step: entering and leaving variables. When choosing entering variable, there may be more than one reduced cost ¯ c j > 0 . When choosing leaving variable, there may be more than one ratio ¯ b i / ¯ a ik that matches the minimum ratio. We may pick entering and leaving variable arbitrarily in the event of multiple choices. On the other hand, we may devise choice rules for choosing entering and leaving variable. We leave the discussion of the choice rules for leaving variable to a later chapter.

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Chapter 6: The Simplex Method 3 Rules for entering variables We discuss four commonly used choice rules for entering variables. We shall use the following tableau ( T ) for illustration. ( T ) z - 2 x 1 - 3 x 2 = 0 x 1 + 2 x 2 + x 3 = 2 x 1 - x 2 + x 4 = 3 Largest coefficient rule. This rule was first suggested by George Dantzig (the inventor of the simplex method). As a result, it is also known as Dantzig’s rule .
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L16 - CO350 Linear Programming Chapter 6 The Simplex Method...

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