Lecture 9: Linear Programming

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Linear Programming ….subject to a series of M constraints N ∑ i= 1 ≤ c ij x i = b j j = 1, 2, 3, …,M ≥ where the b j are all non-negative 195
Linear Programming • The constraints are each represented by N–1 dimensional hyperplanes within the N-dim space – these form a convex hyperpolygon in N dimensions bounding a region of volume V N –if V N is zero (all volume eliminated), there is no solution –if V N is > 0, then the gradient ∇Z = a i e i to a boundary plane ∑ i= 1 leads us • Unless ∇Z is perpendicular to the boundary plane, we can follow that plane to an edge and that edge to a vertex a single optimal vector N 196
Page 1: Lecture 9: Linear Programming • A
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 and 24: Getting the constraints into restri
Page 25 and 26: Getting the constraints into restri

Lecture 9: Linear Programming

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