Lecture 9: Linear Programming

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Getting the constraints into restricted normal form M ∑ First use Z′ = – z j as the “auxiliary” objective function j= 1 And adopt the definitions of the z j as the M constraints. Initially, the z j are all LH variables and the x i are all RH variables. This is a problem in restricted normal form The solution which maximizes Z′ has all the z i equal to zero* after the Simplex method is done, all the z j must become RH variables, and M out of N the x i will be LH variables. *if such a solution exists: if not, then the constraints cannot be satisfied simultaneously 217
Getting the constraints into restricted normal form But now each of the RH variables appears in exactly 1 constraint equation. We zero out all the z j , and have a problem involving the x i in restricted normal form. This we now solve via the Simplex method, using the original objective function N Z = ∑ a i x i i= 1 218
Page 1 and 2: Lecture 9: Linear Programming • A
Page 3 and 4: Linear Programming • The constrai
Page 5 and 6: Feasible vectors • 2-D example: Z
Page 7 and 8: Feasible basic vectors The total nu
Page 9 and 10: The simplex method (illustrated by
Page 11 and 12: Solution via the simplex method Rep
Page 13 and 14: Solution via the simplex method We
Page 15 and 16: Solution via the simplex method Set
Page 17 and 18: Keep pivoting until the “z row”
Page 19 and 20: Obtaining the constraints in restri
Page 21 and 22: Treatment of inequalities (Equivale
Page 23: Getting the constraints into restri

Lecture 9: Linear Programming

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