2011f_OR_I_Chap_05 - Chapter 5 The The Theory of the Simplex Method 5.1 Foundations of The simplex method 1 Geometric solution procedure for

2011f_OR_I_Chap_05 - Chapter 5 The The Theory of the...

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The Theory of the Chapter 5 Simplex Method
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1. Geometric solution procedure ( for 2-dimensional problem) 2. Tableau form solution procedure 3. Matrix form solution procedure & avoid unnecessary computation It streamlines the procedure considerably for computer implementation. Revised Simplex Method 5.1 Foundations of The simplex method Geometric Concept (Geometrical Interpretation) 2 - dimensional dimensional Algebraic Procedure (Algebraic Interpretation) n - dimensional dimensional How thecorner feasible solutionsis identified algebraically?
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Geometric Meaning and Algebraic Procedures: Q: Why do we study thetheory of linearprogramming?
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Starting at vertex ( 0, 0) 18 2 3 12 2 4 0 5 3 5 2 1 4 2 3 1 2 1 x x x x x x x x x z Move to the adjacent vertex (0, 6). Optimal? No
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At vertex (0, 6) 1 4 1 3 2 4 1 4 5 5 3 30 2 4 1 6 2 3 6 z x x x x x x x x x Move to the adjacent vertex (2, 6). Optimal? No
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At vertex (2, 6) 2 3 1 3 1 6 2 1 2 3 1 3 1 36 2 3 5 4 1 4 2 5 4 3 5 4 x x x x x x x x x x z Stop Optimal? Yes
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constraint boundary equations constraint boundary corner-point solution corner-point feasible solution boundary of the feasible region corner-point infeasible solution
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For any LP with n decision variables, each CPF solution is a simultaneous solution of n constraint boundary equations. 1 3 2 4 1 2 5 4 2 12 3 2 18 x x x x x x x
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edge of a feasible region: It lies at the intersection of n 1 constraint equations. Simplex Method: Move from one corner-point feasible solution to an adjacent corner-point feasible solution. Geometric interpretation Algebraic Procedure 1 3 2 4 1 2 5 4 2 12 3 2 18 x x x x x x x Two “adjacent” corner -point feasible solutions: a. Geometric meaning: e.g. (0, 0) and (0, 6), (0, 0) and (4, 0) b. Algebraic meaning : The defining equations. of two “adjacent” corner-point feasible solutions. 1 2 0 (0,0) : 0 x x 1 2 0 (0,6): 2 12 x x 1 2 0 (0,0) : 0 x
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