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lecture14

# lecture14 - Uncapacitated Lot Sizing Activity Selection...

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Uncapacitated Lot Sizing Activity Selection IE170: Algorithms in Systems Engineering: Lecture 14 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University February 23, 2007 Jeff Linderoth IE170:Lecture 14 Uncapacitated Lot Sizing Activity Selection Taking Stock Last Time Lot Sizing and Java Code This Time Lot Sizing—Wagner-Whitin Greedy Algorithm Jeff Linderoth IE170:Lecture 14 Uncapacitated Lot Sizing Activity Selection Uncapacitated Lot Sizing Lot sizing is the canonical production planning problem Given a planning horizon T = { 1 , 2 , . . . , T } You must meet given demands d t for t ∈ T You can meet the demand from a combination of production ( x t ) and inventory ( s t - 1 ) Production cost: c ( x t ) = K + cx t if x t > 0 0 if x t = 0 Inventory cost: I ( s t ) = h t s t Jeff Linderoth IE170:Lecture 14 Uncapacitated Lot Sizing Activity Selection In General A General Recursive Relationship f t ( s ) = min x 0 , 1 , 2 ,... { c t ( x ) + h t ( s + x - d t ) + f t +1 ( s + x - d t ) } . What if K = 250 , d = [220 , 280 , 360 , 140 , 270] , c t = 2 , h t = 1 This might be a problem, as you need to consider producing every possible amount between 0 and 1270 Instead, as is often the case in dynamic programming, we look for structural properties of an optimal solution that will make the algorithm more efficient. Jeff Linderoth IE170:Lecture 14

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Uncapacitated Lot Sizing Activity Selection I Love Lemmas Lemma (Fact) 1 Let x * be an optimal policy (production schedule). If x * t > 0 , then x * t = T - t j =0 d t + j for some j ∈ { 0 , 1 , . . . T - t } Why? Oh Why? If Lemma 1 was false, then there would be some period t and some subsequent period t + j such that production x * t only partially satisfied the demand in t + j . Say this is a quantity 0 < p < d t + j .
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lecture14 - Uncapacitated Lot Sizing Activity Selection...

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