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Evgeny R. Gafarov, Alexander A. Lazarev Institute of Control Sciences of the Russian Academy of Sciences

Moscow, Russia

Frank Werner Otto-von-Guericke-University

Magdeburg, Germany

A Modification of Dynamic Programming Algorithms to Reduce

the Running Time or/and Time Complexity

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Outline of the Talk

1. Dynamic Programming and Graphical Algorithms 2. A Polynomial Algorithm for Problem 1 (nd) | | max ∑ Tj 3. A Numerical Example 4. An Overview of Graphical Algorithms for Single Machine Problems

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1. Dynamic Programming and Graphical Algorithms

Dynamic Programming (Bellman 1957)

Idea of the graphical algorithm: Combine several states into a new state

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Computations in a dynamic programming algorithm

Computations in a graphical algorithm

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2 . A Polynomial Algorithm for Problem 1 (nd) | | max ∑ Tj

no idle times, a schedule starts at time 0

maximization of total tardiness ∑ Tj

single machine n jobs j = 1,2,…,n pj processing time dj due date

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Lemma 1: There exists an optimal schedule π = (G, H), where all jobs in G are on-time and all jobs from H are tardy. All jobs from set G are processed in SPT order and all jobs from set H are processed in LPT order.

Notations: πl(t) – best schedule of jobs 1,2,…,l starting at time t Fl(t) – corresponding total tardiness

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Theorem 1: The graphical algorithm constructs an optimal schedule for problem 1 (nd) | | max ∑ Tj in O(n2) time.

3. A Numerical Example

n = 4

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4. An Overview of Graphical Algorithms for Single Machine Problems

Two Announcements 1. New Book is ready: Sotskov, Sotskova, Lai, Werner: Scheduling under Uncertainty: Theorems and Algorithms, Belarusian Science, Minsk, 326 p., July 2010 (ISBN 978-985-08-1173-8) - free download of a pdf from http://www.math.uni-magdeburg.de/~werner/sot-sot-lai-werner.pdf - paperback book at production cost available 2. LiSA – Version 3.0 with documentation is ready: Andresen, Bräsel, Engelhardt, Werner: LiSA – A Library of Scheduling Algorithms, Handbook for Version 3.0, 107 pages, TR 10-05, August 2010.