Set_1_b_simplex_intro

# Set_1_b_simplex_intro - Dr. Maddah ENMG 500 Engg Management...

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1 Dr. Maddah ENMG 500 Eng’g Management I 10/09/07 Writing LP in standard form In this form all constraints are transformed into equalities. This is achieved by adding slack variables for “≤” constraints and surplus variables for “≥” constraints. For example, the LP 12 max Z 3 4 subject to 3 2 6 4 4 0, 0 xx    is written in standard form as 1 2 1 1 2 2 1 2 1 2 3 4 subject to 3 2 6 4 4 0, 0, 0, 0 x x S x x S S S x x The LP 1 min Z subject to 2 6 2 4 3 0, 0 x  is written in standard form as 1 2 1 1 2 2 13 subject to 2 6 2 4 3 0, 0 ii x x S x x S xS Sx 

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2 Definitions A constraint is binding at a given point if this point is on the constraint (i.e., the point’s coordinates satisfy the constraint as an equality). In a LP with n decision variables, a corner point is a point where n or more constraints are binding. Theorm If a LP has an optimal solution then there exists an optimal solution corresponding to a corner point . Graphical motivation for the simplex method Consider the following LP written in standard form 12 1 2 1 1 2 2 1 2 1 2 max Z 3 4 subject to 3 2 6 4 4 0, 0, 0, 0 xx x x S x x S S S x x  x 1 x 2 2 3 4 1 O A C B(8/5, 3/5) x 2 = 0 x 1 = 0 S 1 = 0 S 2 = 0 nb( x 1 , x 2 ) b( S 1 , S 2 ) nb( x 1 , S 2 ) b( S 1 , x 2 ) nb( S 1 , S 2 ) b( x 1 , x 2 ) Simplex Path D E nb( S 1 , x 2 ) b( x 1 , S 2 ) nb( S 2 , x 2 ) b( x 1 , S 1 ) nb( x 1 , S 1 ) b( S 2 , x 2 )
3 Choose an initial corner point feasible solution (i.e., basic feasible solution). In the current example, O(0, 0) can serve as such solution. At O define the “zero-variables” x 1 and x 2 as the nonbasic variables. Define the remaining nonzero variables S 1 and S 2 as the basic variables. The problem is of the “max” type. To increase the value of Z, one may increase x 1 or x 2 from their current zero levels at O.

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## This note was uploaded on 12/23/2010 for the course ENMG 500 taught by Professor Drbacelmaddah during the Fall '09 term at American University of Beirut.

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Set_1_b_simplex_intro - Dr. Maddah ENMG 500 Engg Management...

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