lecture7-4(Simplex)

# lecture7-4(Simplex) - ISyE Lecture#7 Outline Outline ISyE...

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ISyE 323 Lecture #7 Prof. Jeﬀ Linderoth September 24, 2009 ISyE Lecture #7 Outline Outline The Algebra of the Simplex Method (Section 4.3) The Simplex Method in Tabular Form (Section 4.4) Tie Breaking in the Simplex Method (Section 4.5) Adapting to Other Model Forms (Section 4.6) Postoptimality Analysis (Section 4.7) ISyE Lecture #7 2 ISyE Lecture #7 The Algebra of the Simplex Method (Section 4.3) Outline The Algebra of the Simplex Method (Section 4.3) Introduction Initialization Step 1 Step 2 Step 3 Optimality Test Simplex Iteration: Summary Iteration 2 The Simplex Method in Tabular Form (Section 4.4) Tie Breaking in the Simplex Method (Section 4.5) ISyE Lecture #7 3 ISyE Lecture #7 The Algebra of the Simplex Method (Section 4.3) Introduction Before We Begin. .. ± In this section, we will discuss the algebra of the simplex method. ± We’ll discuss the algebraic equivalent to each of the geometric steps in our geometric explanation. ± We’ll keep the geometry in mind when discussing the algebra. ± But, one important note: ± The geometry is based on the original form of the LP (2 variables). ± The algebra is based on the augmented form of the LP (5 variables). ISyE Lecture #7 4

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ISyE Lecture #7 The Algebra of the Simplex Method (Section 4.3) Initialization Initialization (Step 0) ± In Step 0, we chose (0 , 0) as our initial BF solution (SC #3). ± In other words, we chose x 1 and x 2 as our initial nonbasic variables. ± We need to solve the following system for the basic variables (in red ): x 1 + x 3 = 4 (1) 2 x 2 + x 4 = 12 (2) 3 x 1 + 2 x 2 + x 5 = 18 (3) ± This is easy, since the nonbasic variables equal 0! x 3 = 4 x 4 = 12 x 5 = 18 ± Our BF solution is (0 , 0 , 4 , 12 , 18) . ISyE Lecture #7 5 ISyE Lecture #7 The Algebra of the Simplex Method (Section 4.3) Initialization Optimality Test ± The objective function is Z = 3 x 1 + 5 x 2 so Z = 0 . ± How do we know whether this is optimal? ± If we increased either nonbasic variable ( x 1 or x 2 ), Z would increase. ± Because the coeﬃcients of both x 1 and x 2 are positive in the expression for Z . ± Therefore, (0 , 0 , 4 , 12 , 18) is not optimal. ± For our initial solution, all of the basic variables have coeﬃcients equal to 0 in the expression for Z . ± This will not be true after this iteration. ± So we’ll have to ﬁnd a way of writing Z in terms of just the nonbasic variables. ISyE Lecture #7 6 ISyE Lecture #7 The Algebra of the Simplex Method (Section 4.3) Initialization Summary Summary of Initialization Step ± Choose (0 , 0) as the initial CPF solution if possible. ± Find the corresponding BF solution by solving the constraint system for the basic variables. ± Test for optimality. ISyE Lecture #7 7 ISyE Lecture #7 The Algebra of the Simplex Method (Section 4.3) Step 1 Step 1: Determining the Direction of Movement ± Increasing a nonbasic variable from 0 corresponds to moving along an edge emanating from the current CPF solution.
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lecture7-4(Simplex) - ISyE Lecture#7 Outline Outline ISyE...

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