lect18 - ISE 536Fall03: Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions November 5, 2003 Lecture 18: Large scale LP: Column Generation Lecturer: Fernando Ord´o˜nez 1 Example Consider the problem of operating a supermarket chain. There are a large number of variables that apply only to each store: such as shelf space, shifts scheduling, inventory, cleaning. And there are a number of variables that connect all the stores in the chain: procurement of goods, budget, employee medical benefit negotiation . In the case of two stores this operation can be represented by a linear programming problem such as: ( P ) z = max c t 1 x 1 + c t 2 x 2 s . t . B 1 x 1 + B 2 x 2 = b A 1 x 1 = b 1 A 2 x 2 = b 2 x 1 , x 2 0 The column generation method that we discuss performs better if m << m 1 ,m 2 . 2 Dantzig-Wolfe Decomposition Define S i := { x | A i x = b i , x 0 } , for i = 1 , 2. The problem above is equivalent to ( P ) z = max c t 1 x 1 + c t 2 x 2 s . t . B 1 x 1 + B 2 x 2 = b x 1 S 1 x 2 S 2 Since S i
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lect18 - ISE 536Fall03: Linear Programming and Extensions...

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