Sensitivity analysis and efficient algorithms for some economic lot sizing and scheduling problems [Elektronische Ressource] / vorgelegt von Sergei Chubanov
171 pages
English

Sensitivity analysis and efficient algorithms for some economic lot sizing and scheduling problems [Elektronische Ressource] / vorgelegt von Sergei Chubanov

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171 pages
English
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Sensitivityanalysisande–cientalgorithmsforsomeeconomiclot-sizingandschedulingproblemsDissertationzur Erlangung des akademischen Gradesdoctor rerum politicarum (Dr. rer. pol.)vorgelegt vonDipl.-Inf. Sergei ChubanovSiegen 2006Gutachter: Prof. Dr. Erwin PeschProf. Dr. Peter LetmatheProf. Dr. Mikhail Y. KovalyovTag der mundlic˜ hen Prufung:˜ 30.03.2006Prufer:˜ Prof. Dr. Erwin PeschProf. Dr. Peter LetmatheProf. Dr. Andreas DrexlInternetpublikation der Universit˜atsbibliothek Siegen:urn:nbn:de:hbz:467-2399AcknowledgementsThis thesis presents some parts of my recent research. The thesis would certainly not bewritten without the support of difierent people. I am grateful to all of them. I would liketo thank my supervisor Prof. Dr. Erwin Pesch for giving me the opportunity to work at hischair at the University of Siegen, for his continuing support and advice, and for the patiencewhen reading and commenting the thesis. I thank also Prof. Dr. Letmathe for carefulreading the thesis and for the comments that helped much to improve the thesis text and tocorrect errors. I am very grateful to Prof. Dr. Mikhail Y. Kovalyov, my another supervisorin Minsk, for suggesting interesting directions of research and for all the support. I gratefullyacknowledge Prof. Dr. Mikhail Kovalev, the dean of the faculty of economics of BelarusState University, especially for giving a possibility to work at the chair of MathematicalEconomics. I also thank Dr.

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Publié le 01 janvier 2006
Nombre de lectures 7
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Sensitivityanalysisande–cientalgorithms
forsomeeconomiclot-sizing
andschedulingproblems
Dissertation
zur Erlangung des akademischen Grades
doctor rerum politicarum (Dr. rer. pol.)
vorgelegt von
Dipl.-Inf. Sergei Chubanov
Siegen 2006Gutachter: Prof. Dr. Erwin Pesch
Prof. Dr. Peter Letmathe
Prof. Dr. Mikhail Y. Kovalyov
Tag der mundlic˜ hen Prufung:˜ 30.03.2006
Prufer:˜ Prof. Dr. Erwin Pesch
Prof. Dr. Peter Letmathe
Prof. Dr. Andreas Drexl
Internetpublikation der Universit˜atsbibliothek Siegen:
urn:nbn:de:hbz:467-2399Acknowledgements
This thesis presents some parts of my recent research. The thesis would certainly not be
written without the support of difierent people. I am grateful to all of them. I would like
to thank my supervisor Prof. Dr. Erwin Pesch for giving me the opportunity to work at his
chair at the University of Siegen, for his continuing support and advice, and for the patience
when reading and commenting the thesis. I thank also Prof. Dr. Letmathe for careful
reading the thesis and for the comments that helped much to improve the thesis text and to
correct errors. I am very grateful to Prof. Dr. Mikhail Y. Kovalyov, my another supervisor
in Minsk, for suggesting interesting directions of research and for all the support. I gratefully
acknowledge Prof. Dr. Mikhail Kovalev, the dean of the faculty of economics of Belarus
State University, especially for giving a possibility to work at the chair of Mathematical
Economics. I also thank Dr. Alexander Zaporozhets who supervised my flrst research. My
special thanks to Dipl.-Math. AndreiHorbach and Dr. Yury Nikulin. Discussions with these
persons were always fruitful.
This research was supported by grant 03PEM1SI of the Bundesministerium fur˜ Bildung
und Forschung (BMBF), Germany.Contents
1 Introduction 1
1.1 Preliminary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lot-sizing and scheduling models . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Lot-sizing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Scheduling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Algorithms and complexity . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.5 Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.6 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Bound improvement procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 Bound improvement procedure . . . . . . . . . . . . . . . . . . . . . 21
1.4.2 Modifled bound improvement procedure . . . . . . . . . . . . . . . . 22
2 Single-item capacitated economic lot-sizing problems 25
2.1 Preliminary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 An algorithm for the case with linear costs . . . . . . . . . . . . . . . . . . . 29
2.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.3 Dynamic programming algorithm . . . . . . . . . . . . . . . . . . . . 34
iiiii
2.3 Exponential algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.2 The CELS problem with piecewise linear cost functions . . . . . . . . 44
2.4 A polynomial algorithm for a capacitated economic lot-sizing problem with
piecewise concave cost functions . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5 An FPTAS for a single-item capacitated economic lot-sizing problem with a
monotone cost structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.2 A rounded problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.6 Generalizations of the CELS problem . . . . . . . . . . . . . . . . . . . . . . 81
2.6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.6.2 Non-approximability . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.6.3 Dynamic programming algorithm . . . . . . . . . . . . . . . . . . . . 86
2.6.4 Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . 88
2.6.5 The problem of minimizing maximum cost . . . . . . . . . . . . . . . 96
2.6.6 An FPTAS for the case with product losses . . . . . . . . . . . . . . 97
2.6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Multi-machine scheduling 105
3.1 Preliminary notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2 Local search in large-scale neighborhoods . . . . . . . . . . . . . . . . . . . . 107
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2.2 Exact and approximate search in exponential neighborhoods . . . . . 109iv
3.2.3 Local search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3 An FPTAS for a nonlinear scheduling problem . . . . . . . . . . . . . . . . . 131
3.3.1 Recursive function and its properties . . . . . . . . . . . . . . . . . . 139
3.3.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.3.3 Time-varying machine speeds . . . . . . . . . . . . . . . . . . . . . . 152
3.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Summary 156Chapter 1
Introduction
1.1 Preliminary notes
In many flelds of human activity one faces the necessity of decision making. In production
planning one should decide how many product units to produce in each period, and in
manufacturing one should decide, for instance, how to schedule jobs. In other words, one
shouldselectoneofmanypossiblevariants, ofmanysolutions, andallofthemareofdifierent
quality which can be measured by difierent criteria like cost or time needed to implement a
solution. Obviously, it is reasonable to choose the best one, i.e., an optimal solution. When
we deal with a single quality parameter, say, production cost, which we want to minimize
or maximize, it is usually easy to determine whether one solution is better than another. If
we additionally have a small number of alternatives, we can make an exhaustive search to
determine the best one. It would be an acceptable way, but there is a serious obstacle { time
required to perform the necessary amount of computations. For instance, provided that we
use the exhaustive search, we must explore more than (n¡2)! paths in a complete graph
with arbitrary positive edge weights to flnd a shortest path (i.e., one of minimum weight)
connecting two given nodes. If n = 50; we need billions of years to search all paths even
usingafastestcomputer. Butmostoftenwecannotafiordevenacoupleofdaysjusttoreach
such a modest aim as a solution having minimum cost. Therefore, we need a better way.
For the shortest path problem we have mentioned there is a simple algorithm designed byPreliminary notes 2
2Dijkstra [20] which needs onlyfin arithmetic operations wherefi is a constant. This means
that a common personal computer will need less than a second to flnd a shortest path for
n = 50: The algorithm of Dijkstra is a good illustration both to the concept of polynomial
algorithms and to the fact that sometimes problems that seem to be not solvable in the
period of entire human life require seconds to be solved provided that we use an appropriate
algorithm. This also re ects the main purpose of combinatorial optimization, the branch of
applied mathematics, studying algorithms which are "better than flnite" [23].
Combinatorial optimization problems have deterministic nature, i.e., all problem param-
eters are known in advance, whereas in many real situations necessary information is often
not completely available. However, combinatorial optimization models and methods can
successively be applied even in this case, fore instance, if model parameters such as demand,
costs, job release dates, job processing times, etc. are forecasted. Here difierent methods
from statistics and probability theory may be used to guess these values. After making
a forecast, a model can be treated as deterministic and then corresponding combinatorial
optimization problems are simply formulated using forecast values. Then, having obtained
an optimal or approximately optimal solution, one can use them when planning certain ac-
tivities. Ever

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