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**Unformatted text preview: **ISE 536–Fall03: Linear Programming and Extensions September 24, 2003 Lecture 9: Simplex Method, Degeneracy Lecturer: Fernando Ord´ o˜nez 1 Full simplex algorithm To solve min c t x : Ax = b,x ≥ 0. To start the algorithm we need a basic solution B- 1 b and we represent the problem in Tableau- c t B B- 1 b t c N- c t B B- 1 N B- 1 b I B- 1 A = [ u ij ] i =0 ,...,m j =0 ,...,n . 1. Optimality Test: If ¯ c j ≥ 0 for all j STOP, solution optimal 2. Unboundedness Test: If ¯ c j < 0 and u ij ≤ 0 for all i STOP, unbounded 3. If ¯ c j < 0 and at least one u ij > 0: – x j enters the basis – x l leaves the basis: x l u lj = min u ij > x i u ij – Update the Tableau: u lq = u lq u lj and u iq = u iq- u lq u lj u ij – Goto step 1. 1.1 Starting the Algorithm - Phase I Solve the problem u * = min ∑ m k =1 y k : Ax + Iy = b,x ≥ 0 from (0 ,b ) as the initial BFS using Simplex. • If u * > 0 original problem is infeasible • If u * = 0 and BFS has all y out of the basis B , then solve original problem using B as initial BFS....

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