ISE 203 OR I Chapter 5 Theory of

ISE 203 OR I Chapter 5 Theory of the Simplex Method Asst. Prof. Dr. Nergiz Kasımbeyli

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x 1=0 and x 1=4 x 2=0 and 2 x 2=12 9

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Fig. 5. 3 Not a convex set!

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• Whenever a constraint boundary equation is one of the defining equations for a CP solution, its indicating variable has a value of zero in the augmented form of the problem. • Each such indicating variable is called a nonbasic variable for the corresponding basic solution. 27

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Degenerate solution • A BF solution is a basic solution where all m basic variables are nonnegative (≥ 0). • A BF solution is said to be degenerate if any of these m variables equals zero. • Thus, it is possible for a variable to be zero and still be a basic variable for the current BF solution (Another constraint boundary equation is satisfied in addition to its n defining equations). 29

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The Matrix Form of Simplex 38

The Matrix Form of Simplex 39

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Fundamental Insight Z Decision variables Slack variables rhs All you need to know is B-1 and cb. B-1. You can calculate these from the initial tableau. or You can read them directly off the final tableau.

Fundamental Insight Z Decision variables Slack variables We replace c. BB-1 with y*; and B-1 with S* rhs

Fundamental Insight Z Decision variables Slack variables rhs y* plays a very special role. These are shadow prices. We will often write the final tableau like this. We can use the fundamental insight for sensitivity analysis.

Apply Fundamental Insight • Here is part of the final Tableau for Wyndor • Use the fundamental insight to find the values of the decision variables and the profit.

c. BB-1 b B-1

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Apply Fundamental Insight • Here is part of the final Tableau for Wyndor • Use the fundamental insight to find the values of the decision variables and the profit.

Revised Simplex Method Z Decision variables Slack variables rhs • Find entering basic variable – Calc top row • Find leaving basic variable – Calc rhs and column under entering basic variable • New Solution – Calc new B-1

Revised Simplex A more efficient Simplex Method Instead of computing B-1 in every iteration, it derives B-1 new from B-1 old. 55

Revised Simplex 56

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