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# Analysis of bounds for a capacitated single-item lot-sizing problem.

Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, 1001 West Main Street, P.O. Box 843083, Richmond, VA 23284-3083, USA; School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, NW, Atlanta, GA 30332-0205, USA
Computers & OR 01/2007; 34:1721-1743. DOI:10.1016/j.cor.2005.05.031
Source: DBLP

ABSTRACT Lot-sizing problems are cornerstone optimization problems for production planning with time varying demand. We analyze the quality of bounds, both lower and upper, provided by a range of fast algorithms. Special attention is given to LP-based rounding algorithms.

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##### Article: Approximating the Throughput of Multiple Machines in Real-Time Scheduling
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ABSTRACT: We consider the following fundamental scheduling problem. The input to the problem consists of n jobs and k machines. Each of the jobs is associated with a release time, a deadline, a weight, and a processing time on each of the machines. The goal is to find a schedule that maximizes the weight of jobs that meet their respective deadlines. We give constant factor approximation algorithms for four variants of the problem, depending on the type of the machines (identical vs. unrelated), and the weight of the jobs (identical vs. arbitrary). All these variants are known to be NP-Hard, and the two variants involving unrelated machines are also MAX-SNP hard. The specific results obtained are: -- For identical job weights and unrelated machines: a greedy 2-approximation algorithm. -- For identical job weights and k identical machines: the same greedy algorithm achieves a tight (1+1=k) k (1+1=k) k Gamma1 -approximation factor. ffl For arbitrary job weights and a single machine: an ...
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• ##### Conference Proceeding: Approximating the Throughput of Multiple Machines Under Real-Time Scheduling.
[hide abstract]
ABSTRACT: We consider the following fundamental scheduling problem. The input to the problem consists of n jobs and k machines. Each of the jobs is associated with a release time, a deadline, a weight, anda processing time on each of the machines. The goal is to finda nonpreemptive sched ule that maximizes the weight of jobs that meet their respective deadlines. We give constant factor approximation algorithms for four variants of the problem, depending on the type of the machines (identical vs. unrelated) and the weight of the jobs (identical vs. arbitrary). All these variants are known to be NP-hard , andthe two variants involving unrelatedmachines are also MAX-SNP hard . The specific results obtainedare as follows: • For id entical job weights andunrelatedmachines: a greed y 2-approximation algorithm. • For identical job weights and k identical machines: the same greedy algorithm achieves a tight (1+1/k) k (1+1/k)k−1 approximation factor. • For arbitrary job weights anda single machine: an LP formulation achieves a 2-approximation for polynomially bound edintegral input anda 3-approximation for arbitrary input. For unrelatedmachines, the factors are 3 and4, respectively. • For arbitrary job weights and k id entical machines: the LP-basedalgorithm appliedre- peatedly achieves a (1+1/k) k (1+1/k)k−1 approximation factor for polynomially bounded integral input anda (1+1/2k) k (1+1/2k)k−1 approximation factor for arbitrary input. • For arbitrary job weights andunrelatedmachines: a combinatorial (3 + 2 √ 2 ≈ 5.828)- approximation algorithm.
Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1-4, 1999, Atlanta, Georgia, USA; 01/1999
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##### Article: Scheduling Unrelated Machines by Randomized Rounding
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ABSTRACT: In this paper, we provide a new class of randomized approximation algorithms for parallel machine scheduling problems. The most general model we consider is scheduling unrelated machines with release dates (or even network scheduling) so as to minimize the average weighted completion time. We introduce an LP relaxation in time-indexed variables for this problem. The crucial idea to derive approximation results is not to use standard list scheduling, but rather to assign jobs randomly to machines (by interpreting LP solutions as probabilities), and to perform list scheduling on each of them. Our main result is a (2 + e)--approximation algorithm for this general model which improves upon performance guarantee 16=3 due to Hall, Shmoys, and Wein. In the absence of nontrivial release dates, we get a (3=2 + e)--approximation. At the same time we prove corresponding bounds on the quality of the LP relaxation. A perhaps surprising implication for identical parallel machines is that jo...
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