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dnotes

# dnotes - x s from among the eligible to enter variables say...

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OR320/520 10/25/07 Prof. Bland These pages will be useful for the next 2 classes on duality. See also BHM Chapter 4 scetions 1-4. c 1 c 2 0 0 0 y 1 1 0 0 y 2 A 1 A 2 0 1 0 y 3 0 0 1

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20 30 0 0 0 y 1 2 2 1 0 0 y 2 4 2 0 1 0 y 3 3 6 0 0 1
Recall Part II of the handout Illustrating the Simplex Method . Initial Tableau - z x 1 x 2 x 3 x 4 x 5 1 20 30 0 0 0 0 0 2 2 1 0 0 80 0 4 2 0 1 0 120 0 3 6 0 0 1 210 Second Tableau - z x 1 x 2 x 3 x 4 x 5 1 0 20 0 -5 0 -600 0 0 1 1 - 1 2 0 20 0 1 1 2 0 1 4 0 30 0 0 9 2 0 - 3 4 1 120 You fill in the next Tableau. - z x 1 x 2 x 3 x 4 x 5

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Final Tableau - z x 1 x 2 x 3 x 4 x 5 1 0 0 -5 0 - 10 3 -1100 0 0 1 - 1 2 0 1 3 30 0 1 0 1 0 - 1 3 10 0 0 0 -3 1 2 3 20
A Finite Pivot Selection Rule for the Simplex Method (for maximization). As we will discuss in class, the simplex method, in its general form, does not guarantee termination, because of the possibility of an infinite sequence of consecutive degenerate pivots. We have now seen how to prove the Strong Duality Theorem of linear programming given a finite simplex method. The selection rule below does the job. Most textbooks call it Bland’s Rule . Assume that we are solving a linear programming maximization problem, and B is a feasible basis. Say that nonbasic variable x j is eligible to enter if ¯ c j > 0. Once we have chosen an entering variable
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Unformatted text preview: x s from among the eligible to enter variables, say that basic variable x B i is eligible to leave if ¯ a is > 0, and ¯ b i / ¯ a is = min { ¯ b k / ¯ a ks : ¯ a ks > } . Bland’s Least Index Rule: (CS) Column Selection: select x s , the variable eligible to enter variable having minimum index: s = min { j : ¯ c j > } . (RS) Row Selection: select x B r , the variable eligible to leave variable having minimum index: ¯ a rs > , ¯ b r / ¯ a rs = min { ¯ b i / ¯ a is : ¯ a is > } , and { B r = min { B i : ¯ a is > 0, and ¯ b i / ¯ a is = min { ¯ b k / ¯ a ks : ¯ a ks > }} . Theorem : The simplex method with the least index rule terminates after Fnitely many pivots. This theorem implies the strong duality theorem....
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