L21_10 - Chapter 12 Non-Linear Programming Linear...

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Chapter 12 Non-Linear Programming
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Linear Programming Model 11 22 11 1 12 2 1n n 1 21 1 2 2n n 2 m1 1 2 2 mn n m Maximize ..... subject to a x + a x + . .. +a x b b b nn m cx cx cx x + ++ MM M , ,..., 0 n xx ASSUMPTIONS : Proportionality Assumption – Objective function – Constraints Additivity Assumption – Objective function – Constraints
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What is a non-linear program? • maximize 3 sin x + xy + y 3 - 3z + log z Subject to x 2 + y 2 = 1 x + 4z 2 z 0 • A non-linear program is permitted to have non-linear constraints or objectives. • A linear program is a special case of non- linear programming!
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Nonlinear Programs (NLP) Nonlinear objective function f(x) and/or Nonlinear constraints g i (x) . We will present several examples of non-linear programs. ( ) 12 ,,, ( ) ( ) , 1,2, , n ii Let x x x x Max f x gx b i m =… ≤∀ =
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Unconstrained Facility Location 0 2 4 6 8 10 12 14 16 y 02 4 6 8 1 0 1 21 4 1 6 C (2) (7) B A (19) P ? D (5) x Loc. Dem. A: (8,2) 19 B: (3,10) 7 C: (8,15) 2 D: (14,13) 5 P: ? This is the warehouse location problem with a single warehouse that can be located anywhere in the plane. Distances are “Euclidean.”
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• Costs proportional to distance; known daily demands An NLP 22 82 () xy −+ d(P,A) = 14 13 d(P,D) = minimize 19 d(P,A) + … + 5 d(P,D) subject to: P is unconstrained
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Here are the objective values for 55 different locations. 0 50 100 150 200 250 300 350 values for y Objective value x = 0 x = 2 x = 4 x = 6 x = 8 x = 10 x = 12
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Facility Location. What happens if P must be within a specified region? 0
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L21_10 - Chapter 12 Non-Linear Programming Linear...

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