Lecture7 - Lecture 7: Simplex Method III September 10, 2010...

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Lecture 7: Simplex Method III September 10, 2010
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Moving From One BFS to Another Two adjacent vertices correspond to two BFSs whose basis have m - 1 common basic variables (only one different basic variable). In Example 1: A and B are adjacent vertices, the basis of the corresponding BFSs are ( s 1 ,s 2 ) and ( x 2 ,s 1 ), respectively. s 1 is the common one. When move from one BFS to another one, one variable leaves the basis and one enters the basic set. We need to determine: when to move: optimality conditions how to move: which variable enters the basis and which leaves the basis. 2
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Checking Optimality At a BFS, increase a single nonbasic variable, leave other nonbasic variables unchanged, check if the objective value is improved. (Notice that we need to computer the cor- responding changes in basic variables to preserve equality constraints.) To check the improvement, we can solve the basic variables in terms of the nonbasic variables, and substitute into the ob- jective function. The coefficients of the nonbasic variables in the substituted objective function indicate the improve- ment. If a coefficient is negative (positive) then increasing the cor- responding nonbasic will decrease (increase) the objective value. These coefficients are called the reduced costs . 3
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Where do we go next?
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New direction Gauss-Jordan operations give us expressions for basic variables in terms of nonbasic ones : Suppose we substitute x 1 (basic) in the objective in terms of x 2 and s 1 Z = 20 + 2/3 x 2 -5/6 s 1 This means z increases if x 2 is increased . We can now repeat the steps we used to come to B. Optimality is reached if z can't be increased further (i.e. all coefficients negative) 1x 1 +2/3x 2 + 1/6 s 1 =4 4/3 x 2 -1/6s 1 +1s 2 =2 5/3 x 2 + 1/6 s 1 + 1s 3 =5 1x 2 +1s 4 =2
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Checking Optimality: Examples Back to Example 1: max z = 2 x 1 + 3 x 2 2 x 1 + x 2 + s 1 = 4 x 1 + 2 x 2 + s 2 = 5 x 1 ,x 2 ,s 1 ,s 2 0 , If x 2 and s 1 are basic variables, we can get x 2 = 5 2 - 1 2 x 1 - 1 2 s 2 s 1 = 3 2 - 3 2 x 1 + 1
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This note was uploaded on 09/27/2010 for the course GE 330 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture7 - Lecture 7: Simplex Method III September 10, 2010...

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