Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Nicoló Gusmeroli
Machine Scheduling
to Minimize Weighted
Completion Times
The Use of the α-point
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ó
Nicol Gusmeroli
Machine Scheduling
to Minimize Weighted
Completion Times
a
The Use of the -point
123
NicolóGusmeroli
Institut für Mathematik
Alpen-Adria-Universität Klagenfurt
Klagenfurt, Kärnten
Austria
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-77527-2 ISBN978-3-319-77528-9 (eBook)
https://doi.org/10.1007/978-3-319-77528-9
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To those who ever believed in me…
Preface
This work reviews the most important results regarding the use of the a-point in
SchedulingTheory.ItprovidesanumberofdifferentLPrelaxationsforscheduling
problems and seeks to explain their polyhedral consequences. It also explains
theconceptofthea-pointandhowtheconversionalgorithmworks,pointingoutthe
relationstothesumoftheweightedcompletiontimes.Lastly,thebookexploresthe
latesttechniquesusedformanyschedulingproblemswithdifferentconstraints,such
as release dates, precedences, and parallel machines.
This reference book is intended for advanced undergraduate and postgraduate
studentswhoareinterestedinSchedulingTheory.Itisalsoinspiringforresearchers
wanting to learn about sophisticated techniques and open problems of the field.
Klagenfurt, Austria Nicoló Gusmeroli
January 2018
vii
Contents
Machine Scheduling to Minimize Weighted Completion Times . . . . . . . 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 List of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 LP Relaxations for the Release Dates Case. . . . . . . . . . . . . . . . . . . . . 5
3.1 Time-Indexed Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Mean Busy Time Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Polyhedral Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Conversion Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Some ConsequenquPes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Approximations for 1jrjj wjCj . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.1 2-Approximation Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Another 2-Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . 19
5.3 Bounds for Common a Value . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Bounds for Job-Dependent aj Values . . . . . . . . . . . . . . . . . . . . . 24
5.5 Bad Instances for thPe LP Relaxations. . . . . . . . . . . . . . . . . . . . . 28
6 Approximations for 1jrjj Cj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1 e -Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 28
e(cid:2)1
6.2 Tightness . . . . . . . . . .P. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Approximation for 1jrj;precj wjCj . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.1 LP Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.2 2.54-ApproximationPAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 35
8 Approximations for Pjrjj Cj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.1 3-Approximation Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.2 The DELAY-LIST Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.3 2.83-Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
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9 Approximation for Pjdijj wjCj . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.1 List Scheduling Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.2 4-Approximation Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
10 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References.... .... .... .... ..... .... .... .... .... .... ..... .... 51
About the Author
Nicoló Gusmeroli completed his master’s degree at the ELTE University of
Budapestin2017andiscurrentlyworkingontheprojectHigh-PerformanceSolver
for Binary Quadratic Problems at the Alpen-Adria University of Klagenfurt as a
Ph.D. student. His main research interests are in combinatorial optimization,
semidefinite optimization, and Scheduling Theory.
He completed his bachelor’s studies at the University of Verona prior to
spendinganexchangesemesterattheUniversityofPrimorska(Slovenia),wherehe
wrote his bachelor’s thesis.
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