L03f05

L03f05 - Chapter 1 An Introduction 1 Chek Beng Chua Recap...

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Unformatted text preview: Chapter 1: An Introduction 1 Chek Beng Chua Recap Terminology • Objective function • Constraints • Feasible solution • Feasible region • Objective value • Optimal solution • Optimal value Geometry of LP (Orange Factory Problem)- x 1 6 x 2 A A A A A A @ @ @ @ @ @ @ @ b (0,0) b (0,4) b (1,5) b (4,2) b (5,0) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 2 x 1 + 3 x 2 = 17 More Examples: A transportation problem minimize p i =1 q j =1 c ij x ij subject to q j =1 x ij = s i ( i = 1 , 2 , . . . , p ) p i =1 x ij = t i ( j = 1 , 2 , . . . , q ) x ij ≥ i = 1 , 2 , . . . , p, j = 1 , 2 , . . . , q Chapter 1: An Introduction 2 Chek Beng Chua Some More Examples Overdetermined System of Equations n j =1 a ij x j = b i ( i = 1 , 2 , . . . , m ) Suppose the system does not have a solution. Problem: Find x ∈ R n that is “closest” to solving the system. Error of solution x is m i =1 e i where e i := n j =1 a ij x j- b i Mathematical model minimize m i =1 n j =1 a ij x j- b i Not an LP! Convert to LP minimize m i =1 e i subject to n j =1 a ij x j- b i = e i ( i = 1 , 2 , . . . , m ) Chapter 1: An Introduction 3 Chek Beng Chua...
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L03f05 - Chapter 1 An Introduction 1 Chek Beng Chua Recap...

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