From 92511d35ef4f3182dcd9b06d4363a08ce646f645 Mon Sep 17 00:00:00 2001 From: Timo Date: Tue, 2 May 2023 11:57:06 +0200 Subject: [PATCH 1/4] initial commit with sources --- ads/header.aux | 2 + ads/header.fdb_latexmk | 33 + ads/header.fls | 183 +++ ads/header.log | 223 ++++ bibliographie.bib | 125 +- content/01kapitel.tex | 22 - dokumentation.aux | 93 ++ dokumentation.bbl | 20 + dokumentation.bcf | 2395 +++++++++++++++++++++++++++++++++++++ dokumentation.blg | 9 + dokumentation.fdb_latexmk | 328 +++++ dokumentation.fls | 1960 ++++++++++++++++++++++++++++++ dokumentation.glg | 7 + dokumentation.glo | 0 dokumentation.gls | 6 + dokumentation.ist | 31 + dokumentation.lof | 7 + dokumentation.log | 1603 +++++++++++++++++++++++++ dokumentation.lol | 7 + dokumentation.lot | 7 + dokumentation.run.xml | 88 ++ dokumentation.toc | 12 + sources/sources.bib | 106 ++ 23 files changed, 7220 insertions(+), 47 deletions(-) create mode 100644 ads/header.aux create mode 100644 ads/header.fdb_latexmk create 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(Default extension: tex) + +Enter file name: +d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/ads/header.tex:40: Emergency stop. + + +l.40 \input{ads/einstellungen_liste} + % verfügbare Einstellungen^^M +*** (cannot \read from terminal in nonstop modes) + + +Here is how much of TeX's memory you used: + 3916 strings out of 475075 + 85136 string characters out of 5783001 + 585134 words of memory out of 5000000 + 25354 multiletter control sequences out of 15000+600000 + 470321 words of font info for 30 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 108i,1n,106p,10686b,271s stack positions out of 10000i,1000n,20000p,200000b,200000s +d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/ads/header.tex:40: ==> Fatal error occurred, no output PDF file produced! diff --git a/bibliographie.bib b/bibliographie.bib index ee02bba..036b434 100644 --- a/bibliographie.bib +++ b/bibliographie.bib @@ -1,31 +1,106 @@ -@book{baumgartner:2002, - title = {E-Learning Praxishandbuch : Auswahl von Lernplattformen; Marktübersicht, Funktionen, Fachbegriffe}, - author = {Baumgartner, Peter and Häfele, Hartmut and Maier-Häfele, Kornelia}, - address = {Innsbruck}, - publisher = {StudienVerl.}, - year = {2002}, - isbn = {3-7065-1771-X} +% Books +@book{Luderer2017, + title={Facetten der Wirtschaftsmathematik: Eine unterhaltsame Einf{\"u}hrung ganz ohne Formeln}, + author={Luderer, B.}, + isbn={9783658191870}, + url={https://books.google.de/books?id=F256swEACAAJ}, + year={2017}, + pages = {104-105}, + publisher={Springer Fachmedien Wiesbaden} } -@TechReport{dreyfus:1980, - author = {Stuard E. Dreyfus and Hubert L. Dreyfus}, - title = {A Five-Stage Model Of The Mental Activities Involved In Directed Skill Acquisition}, - institution = {University of California, Berkley}, - year = {1980}, - url = {http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA084551&Location=U2&doc=GetTRDoc.pdf}, +@article{Chu1998, + title={A Genetic Algorithm for the Multidimensional Knapsack Problem}, + author={Chu, P. and Beasley, J.}, + journal={Journal of Heuristics}, + volume={4}, + number={1}, + pages={63-86}, + year={1998}, + doi={10.1023/A:1009642405419}, + publisher={Springer} } -@Online{fsf:2007, - author = {{Free Software Foundation}}, - title = {{GNU AFFERO GENERAL PUBLIC LICENSE}}, - url = {http://www.gnu.org/licenses/agpl-3.0.de.html}, - year = {2007} +@book{Vocking2008, + title={Taschenbuch der Algorithmen}, + editor={Vöcking, B. and weitere}, + year={2008}, + publisher={Springer}, + address={Berlin, Heidelberg}, + pages={405-413}, + isbn={9783540763932} } -@Unpublished{mustermann:2012, - author = {Max Mustermann}, - title = {tolles Musterthema}, - year = {2012}, - address = {Musterstadt}, - note = {Studienarbeit} -} \ No newline at end of file +@incollection{Martello1987, + title = {Algorithms for Knapsack Problems}, + editor = {Silvano Martello and Gilbert Laporte and Michel Minoux and Celso Ribeiro}, + series = {North-Holland Mathematics Studies}, + publisher = {North-Holland}, + volume = {132}, + pages = {213-257}, + year = {1987}, + booktitle = {Surveys in Combinatorial Optimization}, + issn = {0304-0208}, + doi = {https://doi.org/10.1016/S0304-0208(08)73237-7}, + url = {https://www.sciencedirect.com/science/article/pii/S0304020808732377}, + author = {Silvano Martello and Paolo Toth}, + abstract = {Publisher Summary + This chapter presents a great variety of practical problems that are represented by a set of entities, each having an associated value, from which one or more subsets has to be selected in such a way that the sum of the values of the selected entities is maximized and some predefined conditions. The most common condition is obtained by associating a size to each entity and establishing that the sum of the entity sizes in each subset does not exceed some prefixed bound. These problems are generally called knapsack problems, as they recall the situation of a hitch-hiker having to fill up knapsack by selecting from among various possible objects those that provides the maximum comfort. The entities are called as items and their number is indicated by n. The value and size associated with the j-th item are called as profit and weight. The majority of problems are considered as single knapsack problems, where one container must be filled with an optimal subset of items. The chapter reviews both exact and approximate algorithms for the solution of the most important types of knapsack problems. Exact solutions are obtained through branch-and-bound algorithms or dynamic programming, or a combination of the two approaches. The performances of these algorithms are evaluated by computational experiments with randomly generated test problems. Approximate algorithms are of the greedy type or are based on scaling. They are evaluated on the basis of their worst-case performance.} +} + +@INPROCEEDINGS{Lawler1977, + author={Lawler, E. L.}, + booktitle={18th Annual Symposium on Foundations of Computer Science (sfcs 1977)}, + title={Fast approximation algorithms for knapsack problems}, + year={1977}, + volume={}, + number={}, + pages={206-213}, + doi={10.1109/SFCS.1977.11}} + + +@mastersthesis{Schury2013, + title={Social Choice Methoden zur Lösung von mehrdimensionalen Rucksackproblemen / Social Choice Methods for the solution of the multimensional knapsack problem}, + author={Schury, Christoph}, + year={2013}, + school={Universität Graz}, + language={German}, + keywords={Rucksackproblem, Kollektiventscheidung, Online-Publikation}, + urn={urn:nbn:at:at-ubg:1-47652}, + pages={IV, 55}, + advisor={Pferschy, Ulrich} +} + + +@Inbook{Domschke2015, +author="Domschke, Wolfgang +and Drexl, Andreas +and Klein, Robert +and Scholl, Armin", +title="Ganzzahlige und kombinatorische Optimierung", +bookTitle="Einf{\"u}hrung in Operations Research", +year="2015", +publisher="Springer Berlin Heidelberg", +address="Berlin, Heidelberg", +pages="127--164", +abstract="Die meisten der in den Kapiteln 2 bis 5 betrachteten Probleme sind als lineare Optimierungsprobleme formulierbar und mit dem Simplex-Algorithmus oder spezialisierten Vorgehensweisen l{\"o}sbar. Wesentliche Eigenschaft dieser Probleme ist neben der Linearit{\"a}t von Zielfunktion und Nebenbedingungen, dass ausschlie{\ss}lich kontinuierliche Variablen vorkommen. Im Gegensatz dazu wenden wir uns nun einer Klasse von Problemen zu, bei der auch bin{\"a}re oder ganzzahlige Variablen zugelassen sind. Speziell durch die Verwendung von Bin{\"a}rvariablen ergibt sich die M{\"o}glichkeit, auch komplexere Wirkungszusammenh{\"a}nge wie etwa logische Bedingungen zu modellieren. So beruht die Formulierung des Grundmodells der Kapazit{\"a}tsplanung im Projektmanagement (vgl. Kap. 5.5) auf dem Einsatz von Bin{\"a}rvariablen. Weitere Einsatzgebiete der ganzzahligen und kombinatorischen Optimierung umfassen die Ablaufplanung, die Logistik und die Personaleinsatzplanung (vgl. auch die Beispiele in Kap. 1.4).", +isbn="978-3-662-48216-2", +doi="10.1007/978-3-662-48216-2_6", +url="https://doi.org/10.1007/978-3-662-48216-2_6" +} + + +@article{Martello1990, +title = {An exact algorithm for large unbounded knapsack problems}, +journal = {Operations Research Letters}, +volume = {9}, +number = {1}, +pages = {15-20}, +year = {1990}, +issn = {0167-6377}, +doi = {https://doi.org/10.1016/0167-6377(90)90035-4}, +url = {https://www.sciencedirect.com/science/article/pii/0167637790900354}, +author = {Silvano Martello and Paolo Toth}, +keywords = {integer programming, knapsack problem, branch-and-bound}, +abstract = {Given n types, each having an associated profit and weight, and a container of given capacity, the unbounded knapsack problem is to determine the number of items of each type to be selected so that the corresponding total profit is a maximum and the corresponding total weight does not exceed the capacity. We present upper bounds, dominance relations, and an approach—based on the definition of a core problem—to the exact solution of very large instances of the problem. We give the results of computational experiments on randomly generated test problems involving up to 250 000 item types.} +} diff --git a/content/01kapitel.tex b/content/01kapitel.tex index afb2721..45904e4 100644 --- a/content/01kapitel.tex +++ b/content/01kapitel.tex @@ -1,25 +1,3 @@ %!TEX root = ../dokumentation.tex \chapter{Das erste Kapitel} -Erste Erwähnung eines Akronyms wird als Fußnote angezeigt. Jede weitere wird -nur verlinkt: \acf{AGPL}. \cite{fsf:2007} - -Verweise auf das Glossar: \gls{Glossareintrag}, \glspl{Glossareintrag} - -Nur erwähnte Literaturverweise werden auch im Literaturverzeichnis gedruckt: -\cite{baumgartner:2002}, \cite{dreyfus:1980} - -Meine erste Fußnote\footnote{Ich bin eine Fußnote} - -\begin{wrapfigure}{r}{.4\textwidth} -\centering -\includegraphics[height=.35\textwidth]{logo.png} -\vspace{-15pt} -\caption{Das Logo der Musterfirma\footnotemark} -\end{wrapfigure} -%Quelle muss in Fußnote stehen (da sonst aufgrund eines Fehlers nicht kompiliert -% wird) -\footnotetext{aus \cite{mustermann:2012}} -Looking for the one superhero comic you just have to read. Following the antics and adventures of May Mayday Parker, this Spider-book has everything you could want in a comic--action, laughs, mystery and someone in a Spidey suit. Collects Alias \#1-28, What If. Jessica Jones had Joined the Avengers. In her inaugural arc, Jessicas life immediately becomes expendable when she uncovers the potentially explosive secret of one heros true identity. In her inaugural arc, Jessicas life immediately becomes expendable when she uncovers the potentially explosive secret of one heros true identity. - -Once upon a time, Jessica Jones was a costumed super-hero, just not a very good one. First, a story where Wolverine and Hulk come together, and then Captain America and Cable meet up. In a city of Marvels, Jessica Jones never found her niche. The classic adventures of Spider-Man from the early days up until the 90s. Looking for the one superhero comic you just have to read. 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+ biblatex.cfg + german.lbx + ngerman.lbx + english.lbx + + + + biber + + biber + dokumentation + + + dokumentation.bcf + + + dokumentation.bbl + + + dokumentation.bbl + + + dokumentation.bcf + + + bibliographie.bib + + + diff --git a/dokumentation.toc b/dokumentation.toc new file mode 100644 index 0000000..fd44825 --- /dev/null +++ b/dokumentation.toc @@ -0,0 +1,12 @@ +\babel@toc {ngerman}{}\relax +\contentsline {chapter}{\nonumberline Abkürzungsverzeichnis}{V}{chapter*.4}% +\contentsline {chapter}{\nonumberline Abbildungsverzeichnis}{VI}{chapter*.5}% +\contentsline {chapter}{\nonumberline Tabellenverzeichnis}{VII}{chapter*.6}% +\contentsline {chapter}{\nonumberline Listings}{VIII}{chapter*.7}% +\contentsline {chapter}{\numberline {1}Das erste Kapitel}{1}{chapter.1}% +\contentsline {chapter}{\numberline {2}Beispiel Code-schnipsel einbinden}{2}{chapter.2}% +\contentsline {section}{\numberline {2.1}lorem ipsum}{2}{section.2.1}% +\contentsline {section}{\numberline {2.2}Verweis auf Code}{3}{section.2.2}% +\contentsline {chapter}{Glossar}{4}{section*.9}% +\contentsline {chapter}{\nonumberline Anhang}{5}{chapter*.11}% +\providecommand \tocbasic@end@toc@file {}\tocbasic@end@toc@file diff --git a/sources/sources.bib b/sources/sources.bib new file mode 100644 index 0000000..036b434 --- /dev/null +++ b/sources/sources.bib @@ -0,0 +1,106 @@ +% Books +@book{Luderer2017, + title={Facetten der Wirtschaftsmathematik: Eine unterhaltsame Einf{\"u}hrung ganz ohne Formeln}, + author={Luderer, B.}, + isbn={9783658191870}, + url={https://books.google.de/books?id=F256swEACAAJ}, + year={2017}, + pages = {104-105}, + publisher={Springer Fachmedien Wiesbaden} +} + +@article{Chu1998, + title={A Genetic Algorithm for the Multidimensional Knapsack Problem}, + author={Chu, P. and Beasley, J.}, + journal={Journal of Heuristics}, + volume={4}, + number={1}, + pages={63-86}, + year={1998}, + doi={10.1023/A:1009642405419}, + publisher={Springer} +} + +@book{Vocking2008, + title={Taschenbuch der Algorithmen}, + editor={Vöcking, B. and weitere}, + year={2008}, + publisher={Springer}, + address={Berlin, Heidelberg}, + pages={405-413}, + isbn={9783540763932} +} + +@incollection{Martello1987, + title = {Algorithms for Knapsack Problems}, + editor = {Silvano Martello and Gilbert Laporte and Michel Minoux and Celso Ribeiro}, + series = {North-Holland Mathematics Studies}, + publisher = {North-Holland}, + volume = {132}, + pages = {213-257}, + year = {1987}, + booktitle = {Surveys in Combinatorial Optimization}, + issn = {0304-0208}, + doi = {https://doi.org/10.1016/S0304-0208(08)73237-7}, + url = {https://www.sciencedirect.com/science/article/pii/S0304020808732377}, + author = {Silvano Martello and Paolo Toth}, + abstract = {Publisher Summary + This chapter presents a great variety of practical problems that are represented by a set of entities, each having an associated value, from which one or more subsets has to be selected in such a way that the sum of the values of the selected entities is maximized and some predefined conditions. The most common condition is obtained by associating a size to each entity and establishing that the sum of the entity sizes in each subset does not exceed some prefixed bound. These problems are generally called knapsack problems, as they recall the situation of a hitch-hiker having to fill up knapsack by selecting from among various possible objects those that provides the maximum comfort. The entities are called as items and their number is indicated by n. The value and size associated with the j-th item are called as profit and weight. The majority of problems are considered as single knapsack problems, where one container must be filled with an optimal subset of items. The chapter reviews both exact and approximate algorithms for the solution of the most important types of knapsack problems. Exact solutions are obtained through branch-and-bound algorithms or dynamic programming, or a combination of the two approaches. The performances of these algorithms are evaluated by computational experiments with randomly generated test problems. Approximate algorithms are of the greedy type or are based on scaling. They are evaluated on the basis of their worst-case performance.} +} + +@INPROCEEDINGS{Lawler1977, + author={Lawler, E. L.}, + booktitle={18th Annual Symposium on Foundations of Computer Science (sfcs 1977)}, + title={Fast approximation algorithms for knapsack problems}, + year={1977}, + volume={}, + number={}, + pages={206-213}, + doi={10.1109/SFCS.1977.11}} + + +@mastersthesis{Schury2013, + title={Social Choice Methoden zur Lösung von mehrdimensionalen Rucksackproblemen / Social Choice Methods for the solution of the multimensional knapsack problem}, + author={Schury, Christoph}, + year={2013}, + school={Universität Graz}, + language={German}, + keywords={Rucksackproblem, Kollektiventscheidung, Online-Publikation}, + urn={urn:nbn:at:at-ubg:1-47652}, + pages={IV, 55}, + advisor={Pferschy, Ulrich} +} + + +@Inbook{Domschke2015, +author="Domschke, Wolfgang +and Drexl, Andreas +and Klein, Robert +and Scholl, Armin", +title="Ganzzahlige und kombinatorische Optimierung", +bookTitle="Einf{\"u}hrung in Operations Research", +year="2015", +publisher="Springer Berlin Heidelberg", +address="Berlin, Heidelberg", +pages="127--164", +abstract="Die meisten der in den Kapiteln 2 bis 5 betrachteten Probleme sind als lineare Optimierungsprobleme formulierbar und mit dem Simplex-Algorithmus oder spezialisierten Vorgehensweisen l{\"o}sbar. Wesentliche Eigenschaft dieser Probleme ist neben der Linearit{\"a}t von Zielfunktion und Nebenbedingungen, dass ausschlie{\ss}lich kontinuierliche Variablen vorkommen. Im Gegensatz dazu wenden wir uns nun einer Klasse von Problemen zu, bei der auch bin{\"a}re oder ganzzahlige Variablen zugelassen sind. Speziell durch die Verwendung von Bin{\"a}rvariablen ergibt sich die M{\"o}glichkeit, auch komplexere Wirkungszusammenh{\"a}nge wie etwa logische Bedingungen zu modellieren. So beruht die Formulierung des Grundmodells der Kapazit{\"a}tsplanung im Projektmanagement (vgl. Kap. 5.5) auf dem Einsatz von Bin{\"a}rvariablen. Weitere Einsatzgebiete der ganzzahligen und kombinatorischen Optimierung umfassen die Ablaufplanung, die Logistik und die Personaleinsatzplanung (vgl. auch die Beispiele in Kap. 1.4).", +isbn="978-3-662-48216-2", +doi="10.1007/978-3-662-48216-2_6", +url="https://doi.org/10.1007/978-3-662-48216-2_6" +} + + +@article{Martello1990, +title = {An exact algorithm for large unbounded knapsack problems}, +journal = {Operations Research Letters}, +volume = {9}, +number = {1}, +pages = {15-20}, +year = {1990}, +issn = {0167-6377}, +doi = {https://doi.org/10.1016/0167-6377(90)90035-4}, +url = {https://www.sciencedirect.com/science/article/pii/0167637790900354}, +author = {Silvano Martello and Paolo Toth}, +keywords = {integer programming, knapsack problem, branch-and-bound}, +abstract = {Given n types, each having an associated profit and weight, and a container of given capacity, the unbounded knapsack problem is to determine the number of items of each type to be selected so that the corresponding total profit is a maximum and the corresponding total weight does not exceed the capacity. We present upper bounds, dominance relations, and an approach—based on the definition of a core problem—to the exact solution of very large instances of the problem. We give the results of computational experiments on randomly generated test problems involving up to 250 000 item types.} +} From adaa67e4ea1a26b4a9ef2695b0d934e818bfaa0d Mon Sep 17 00:00:00 2001 From: Timo Date: Tue, 2 May 2023 12:09:16 +0200 Subject: [PATCH 2/4] set variables --- ads/deckblatt.tex | 2 +- dokumentation.aux | 34 ++++++------ dokumentation.fdb_latexmk | 36 +++++------- dokumentation.fls | 53 ++++-------------- dokumentation.log | 114 +++++++++++++------------------------- dokumentation.tex | 18 +++--- dokumentation.toc | 12 ++-- einstellungen.tex | 30 +++++----- 8 files changed, 111 insertions(+), 188 deletions(-) diff --git a/ads/deckblatt.tex b/ads/deckblatt.tex index 9be00d3..1f2d73f 100644 --- a/ads/deckblatt.tex +++ b/ads/deckblatt.tex @@ -2,7 +2,7 @@ \begin{titlepage} \begin{longtable}{p{8.2cm} p{5.4cm}} - {\raisebox{\ht\strutbox-\totalheight}{\includegraphics{images/firma-deckblatt.png}}} & + % 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Christian Kratochwil} +\setzegutachter{} +\setzezeitraum{April 2023 - Juni 2023} +\setzearbeit{Hausarbeit} +\setzeautor{Timo Feucht, Alwin Zomotor} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% Literaturverzeichnis %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From 72017b090c9be35ea1110a4b7a4b521d5acce6cc Mon Sep 17 00:00:00 2001 From: Timo Date: Tue, 2 May 2023 13:15:09 +0200 Subject: [PATCH 3/4] Einleitung initial commit --- content/01kapitel.tex | 52 +++++++++++++++++++++++- dokumentation.aux | 48 +++++++++++++--------- dokumentation.blg | 18 ++++----- dokumentation.fdb_latexmk | 33 +++++++++------ dokumentation.fls | 12 +++++- dokumentation.log | 85 ++++++++++++++++++++++----------------- dokumentation.lol | 4 +- dokumentation.run.xml | 4 +- dokumentation.toc | 18 ++++++--- 9 files changed, 183 insertions(+), 91 deletions(-) diff --git a/content/01kapitel.tex b/content/01kapitel.tex index 45904e4..6f0bfe9 100644 --- a/content/01kapitel.tex +++ b/content/01kapitel.tex @@ -1,3 +1,53 @@ %!TEX root = ../dokumentation.tex -\chapter{Das erste Kapitel} +\chapter{Einleitung} +Bei dem Rucksackproblem handelt es sich um eine einfach zu beschreibende, aber auf keinen Fall einfach zu lösende Aufgabe der diskreten Mathematik. Zur Lösung des Problems werden meist heuristische Verfahren oder Methoden der dynamischen Optimierung verwendet. (Vgl. Luderer, S.104) +Anschaulich geht es bei dem Rucksackproblem um einen Rucksack, der höchstens ein bestimmtes vorgegebenes Gewicht $T$ tragen kann. $T$ wird als Gewichtsschranke bezeichnet. Neben dem Rucksack gibt es eine Menge von Gegenschänden. Diese werden als n Objekten bezeichnet. Jedes Objekt hat ein Gewicht und einen Profit. Untersucht wird nun die Teilmengen der Objekte, die die Gewichtsschranke $T$ nicht verletzten, dabei aber einen möglichst großen Gesamtprofit haben. Das heißt, die Summe der Profite der eingepackten Objekte maximal wird. +Rein intuitiv erscheint es sinnvoll, die Objekte mit dem größten Profit pro Gewichtseinheit zuerst auszuwählen. Das Verhältnis zwischen Profit und Gewicht eines Objektes wird auch als Profitdichte bezeichnet. Die intuitive Lösung ist allerdings nicht immer die beste, da die Gewichtsschranke $T$ teilweise stark unterschritten wird, ein weiteres Objekt jedoch die Gewichtsschranke überschreiten würde. Das heißt, es wird möglicherweise Platz verschwendet. +Die beste Lösung kann garantiert durch Ausprobieren aller möglichen Kombinationen gefunden werden. Allerdings gibt es sehr viele Kombinationsmöglichkeiten. Für jedes einzelne Objekt ist zu entscheiden, ob es in den Rucksack gepackt wird oder nicht. Somit gibt es pro Objekt zwei Möglichkeiten. Bei $n$ Objekten ergeben sich $2^n$ Kombinationsmöglichkeiten. Durch das exponentielle Wachstum der Kombinationsmöglichkeiten „explodiert“ die Anzahl der möglichen Teilmengen. Beispielsweise wäre das Rucksackproblem bei 60 Objekten durch diesen Ansatz mit heutigen Mitteln kaum lösbar, da es bereits $2^60 = 1 152 921 504 606 846 976$ Kombinationsmöglichkeiten gibt. Angenommen, ein Computer schafft es, eine Milliarde Teilmengen pro Sekunden zu berechnen und zu testen. Dann benötigte dieser trotzdem noch über 36 Jahre, um alle Teilmengen auszuprobieren. (vgl. Beier, S.405ff) + +In dieser Arbeit wird auf das klassische binäre Rucksackproblem, auch 0-1 Rucksackproblem genannt, eingegangen. Dabei gibt es nur die Möglichkeit ein Objekt einzupacken oder nicht. Kopien eines Objektes sind dabei nicht erlaubt. Varianten des Rucksackproblems, in denen Kopien eines Objektes erlaubt sind, wie zum Beispiel das Bounded Rucksackproblem, werden in wenigen Sätzen erläutert. + +\section{Mathematische Formulierung} +(Vgl. Matello, S.214) +Mathematisch kann das Rucksackproblem durch ein Ganzzahliges Lineares Programm abgebildete werden. Dabei gebe es ein Rucksack oder anderes Gefäß mit Kapazität $c$ und $n$ Objekt beziehungsweise Gegenstände mit + Profit $p_j,\ \ \ j=1,\ldots,n$ und + Gewicht $w_j,\ \ j=1,\ldots,n$. +Die einfachste Form des Rucksackproblems, das 0-1 Rucksackproblem, kann mathematisch folgendermaßen dargestellt werden: + +\begin{eqnarray} + Maximize + z=\sum_{j=1}^{n}{p_jx_j} + Subject to + \sum_{j=1}^{n}{w_jx_j\le c} + x_j\in\{0,1\},\ \ j=1,\ldots,n +\end{eqnarray} + +Dabei ist $x_j = 1$, wenn das Objekt an der Stelle $j$ ausgewählt wird, wird das Objekt nicht ausgewählt ist $x_j = 0$. Da Profite und Gewichte positiv sind, wird ohne Beschränkung der Allgemeinheit angenommen, dass gilt: +\begin{eqnarray} + \sum_{j=1}^{n}{w_j>c} + w_j\le c,\ \ j=1,\ldots,n +\end{eqnarray} + + +\section{Komplexität NP-Schwer/Vollständig} +(Vgl. Martello, S.233) +Das Rucksackproblem ist NP-Schwer und NP-Vollständig, weil bewiesen wurde, dass das Teilsummenproblem (Subset-Sum Problem SSB) ebenfalls NP-Schwer und NP-Vollständig ist. Beweis aufführen? vgl. Martello S.233f Das Teilsummenproblem ein Spezialfall des Rucksackproblems unter der Bedingung, dass $p_j=w_j$ für alle $j$. Das Teilsummenproblem kann mathematisch wie folgt dargestellt werden: +\begin{eqnarray} +maximize +z=\sum_{j=1}^{n}{w_jx_j} +Subject to +\sum_{j=1}^{n}{w_jx_j\le c} +x_j\in0,1,\ \ j=1,\ldots,n +\end{eqnarray} + +Dadurch kann das Rucksackproblem wiederum auf das Teilsummenproblem reduziert werden. + +\section{verschiedene Arten des Rucksackproblems} +Neben dem klassischen 0-1 Rucksackproblem gibt es weitere Formen des Problems, die im Folgenden in einigen Sätzen beschrieben werden. Es werden nicht alle Varianten des Rucksackproblems aufgeführt. +\subsection{Bounded Rucksackproblem} +Beim Bounded Rucksackproblemen können endlich viele identische Kopien $b_j$ eines Objektes $j$ eingepackt werden. Dadurch ändert sich lediglich die Bedingung der Formel Referenz einfügen von $x_j\in\{0,1\}$ zu $0\le\ x_j\le\ b_j$. Dadurch kann $x_j$ jeden ganzzahligen Wert $0$ und $b_j$ annehmen. Vgl. Martello1990, S.81f Das Unbounded Rucksackproblem kann in das klassische 0-1 Rucksackproblem transformiert werden vgl. Schury, S.17. +\subsection{Unbounded Rucksackproblem} +Beim Unbounded Rucksackproblem können im Gegensatz zum Bounded Rucksackproblem unendlich viele identische Kopien $b_j$ eines Objektes $j$ eingepackt werden. Dadurch ändert sich die Bedingung der Formel Referenz einfügen von $x_j\in\{0,1\}$ zu $x_j\geq\ 0$. Dadurch kann $x_j$ jeden ganzzahligen Wert größer als $0$ annehmen. Das Unbounded Rucksackproblem kann in das 0-1 Rucksackproblem transformiert werden. vgl. Schury, S.17 +\subsection{Multiples Rucksackproblem} +Bei dem Multiplen Rucksackproblem stehen anstatt eines einzelnen Rucksacks oder Gefäß $m$ Rucksäcke mit unterschiedlichen Kapazitäten $c_m$ zur Verfügung. Alle Gegenstände sollen nun in die Rucksäcke verteilt werden, sodass keine Gewichtsschranke überschritten und der Profit maximal wird. Dabei wird aus der Binärvariable $x_j$ des 0-1 Rucksackproblems eine Binärvariablenmatrix. Diese beschreibt, welches Objekt $j$ in welchen Rucksack $i$ eingepackt wird. (vgl. 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\@writefile{lof}{\providecommand\tocbasic@end@toc@file{}\tocbasic@end@toc@file} \@writefile{lot}{\providecommand\tocbasic@end@toc@file{}\tocbasic@end@toc@file} -\gdef \@abspage@last{12} +\gdef \@abspage@last{14} diff --git a/dokumentation.blg b/dokumentation.blg index 87c1309..f8e7cf0 100644 --- a/dokumentation.blg +++ b/dokumentation.blg @@ -1,9 +1,9 @@ -[0] Config.pm:306> INFO - This is Biber 2.18 -[0] Config.pm:309> INFO - Logfile is 'dokumentation.blg' -[246] biber-MSWIN32:340> INFO - === -[319] Biber.pm:418> INFO - Reading 'dokumentation.bcf' -[803] Biber.pm:978> INFO - Found 0 citekeys in bib section 0 -[866] bbl.pm:654> INFO - Writing 'dokumentation.bbl' with encoding 'UTF-8' -[867] bbl.pm:757> INFO - Output to dokumentation.bbl -[869] Biber.pm:130> WARN - The file 'dokumentation.bcf' does not contain any citations! -[870] Biber.pm:132> INFO - WARNINGS: 1 +[4] Config.pm:306> INFO - This is Biber 2.18 +[5] Config.pm:309> INFO - Logfile is 'dokumentation.blg' +[1605] biber-MSWIN32:340> INFO - === +[2117] Biber.pm:418> INFO - Reading 'dokumentation.bcf' +[3780] Biber.pm:978> INFO - Found 0 citekeys in bib section 0 +[4007] bbl.pm:654> INFO - Writing 'dokumentation.bbl' with encoding 'UTF-8' +[4009] bbl.pm:757> INFO - Output to dokumentation.bbl +[4013] Biber.pm:130> WARN - The file 'dokumentation.bcf' does not contain any citations! +[4014] Biber.pm:132> INFO - WARNINGS: 1 diff --git a/dokumentation.fdb_latexmk b/dokumentation.fdb_latexmk index fc88f3d..d0d09d3 100644 --- a/dokumentation.fdb_latexmk +++ b/dokumentation.fdb_latexmk @@ -1,14 +1,14 @@ # Fdb version 3 -["biber dokumentation"] 1683021355 "dokumentation.bcf" "dokumentation.bbl" "dokumentation" 1683022092 - "dokumentation.bcf" 1683022091 107388 34346f1d15652dcd41bb57252a5d6744 "pdflatex" +["biber dokumentation"] 1683025888 "dokumentation.bcf" "dokumentation.bbl" "dokumentation" 1683026027 + "dokumentation.bcf" 1683026022 107388 34346f1d15652dcd41bb57252a5d6744 "pdflatex" (generated) "dokumentation.bbl" "dokumentation.blg" -["cusdep glo gls dokumentation"] 1683021244 "dokumentation.glo" "dokumentation.gls" "dokumentation" 1683022092 - "dokumentation.glo" 1683021272 206 6080184c265691907ecc7081c77d2a97 "pdflatex" +["cusdep glo gls dokumentation"] 0 "dokumentation.glo" "dokumentation.gls" "dokumentation" 1683026027 + "dokumentation.glo" 0 -1 0 "pdflatex" (generated) "dokumentation.gls" -["pdflatex"] 1683022083 "d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/dokumentation.tex" "dokumentation.pdf" "dokumentation" 1683022092 +["pdflatex"] 1683025965 "d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/dokumentation.tex" "dokumentation.pdf" "dokumentation" 1683026027 "ads/acronyms.tex" 1683020997 1196 22948ccac06f5845c6dbe531485c5ebb "" "ads/appendix.tex" 1683020997 2300 d497f5c07b158c2a1bfdff0eb60ee0f4 "" "ads/deckblatt.tex" 1683021923 1219 48d2435b494a85b5b4f5ffacb2542253 "" @@ -16,7 +16,10 @@ "ads/glossary.tex" 1683020997 415 8fcc3eec5a4a302cc681b41d86b74292 "" "ads/header.tex" 1683020997 6116 9198c1e251ab3f6b1a06bf6bb3664357 "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-ec.enc" 1664476421 2375 baa924870cfb487815765f9094cf3728 "" + "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc" 1664476421 3486 c7eadf5dcc57b3b2d11736679f6636ba "" + "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc" 1664476421 2405 5dcf2c1b967ee25cc46c58cd52244aed "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc" 1664476421 2840 216e6e45ad352e2456e1149f28885bee "" + "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc" 1664476421 2327 9d6df24f9c4f7368395224341a95523a "" "c:/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1664466751 3524 cb3e574dea2d1052e39280babc910dc8 "" "c:/texlive/2022/texmf-dist/fonts/tfm/jknappen/ec/ecrm1200.tfm" 1664466279 3584 f80ddd985bd00e29e9a6047ebd9d4781 "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1664465627 1288 655e228510b4c2a1abe905c368440826 "" @@ -43,10 +46,14 @@ "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/rm-lmr6.tfm" 1664476421 11836 e3b6ce3e601aec94f64a536e7f4224d5 "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/rm-lmr8.tfm" 1664476421 11864 309fd7f43e4a0ba39f6f7644d76e8edf "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmbx12.pfb" 1664476421 116908 9a7a12d1e4df280ea0ab4aad83cdef88 "" + "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmex10.pfb" 1664476421 23055 2e5b42921de910eaa97b85df04ca4891 "" + "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmmi12.pfb" 1664476421 30696 3da7e46ecf5615eedebd9f27dab4d9d7 "" + "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmmi8.pfb" 1664476421 30635 833ec815d446ec453a4913fc26d24cbc "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr10.pfb" 1664476421 119235 05e5c8d6d60439433e50047898c2c0e0 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr12.pfb" 1664476421 113634 9384aebdf26f9881e046a4b5574566db "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr17.pfb" 1664476421 119752 1bd8d06e4079df624bf59ce3ad7c9aa6 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr6.pfb" 1664476421 123394 992f5e87d698d9e94717394c62580ea6 "" + "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr8.pfb" 1664476421 122174 a2d5e73f396ddbeb3bdf5c79aea0993d "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmri12.pfb" 1664476421 109265 e6c7d377db6dfccd4a123fb26da8cffb "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmri8.pfb" 1664476421 109952 907d11d9707997c3fd76af7ec651ec78 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmro12.pfb" 1664476421 125209 a5bd29c5b5feb791d0332220e4a79275 "" @@ -290,18 +297,18 @@ "c:/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1664479675 4635069 7677ee69604aae14ed48b153c111a7f7 "" "c:/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1664479754 3058675 f649c66cce6a93d69f7b71a9c0c74319 "" "c:/texlive/2022/texmf.cnf" 1664479650 713 e69b156964470283e0530f5060668171 "" - "content/01kapitel.tex" 1683021342 66 49e907fc3f52751796c0837d57e5c2e4 "" + "content/01kapitel.tex" 1683025841 6045 61e8239feaa6390abc455d3e604e8feb "" "content/02kapitel.tex" 1683020997 2961 0b3ff97ffcf1a769e39c4fc613c93d20 "" "d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/dokumentation.tex" 1683022073 2141 41b8bc6d0729e593fa006ed0b61ac698 "" - "dokumentation.aux" 1683022091 6888 6d00934f574a75f1083f3a90ee6337e6 "pdflatex" - "dokumentation.bbl" 1683021359 486 ee04f40f768daeed26442158e95aba61 "biber dokumentation" + "dokumentation.aux" 1683026022 8983 f1ec91fd0443f5d436f4ca38aadfa1fb "pdflatex" + "dokumentation.bbl" 1683025911 486 ee04f40f768daeed26442158e95aba61 "biber dokumentation" "dokumentation.gls" 1683021246 294 52680d5d082eb5cc2632c73527e50283 "cusdep glo gls dokumentation" - "dokumentation.lof" 1683022091 195 6d2221321a29ec9cab7237bd5c68de69 "pdflatex" - "dokumentation.lol" 1683022091 256 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a/dokumentation.fls b/dokumentation.fls index 6d637af..b462793 100644 --- a/dokumentation.fls +++ b/dokumentation.fls @@ -1871,6 +1871,9 @@ INPUT content/01kapitel.tex INPUT ./content/01kapitel.tex INPUT ./content/01kapitel.tex INPUT content/01kapitel.tex +INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmssbx10.tfm +INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmssbx10.tfm +INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmro12.tfm INPUT ./content/02kapitel.tex INPUT ./content/02kapitel.tex INPUT ./content/02kapitel.tex @@ -1891,8 +1894,6 @@ INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmtt10.tfm INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmr10.tfm INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmr6.tfm INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmr10.tfm -INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmssbx10.tfm -INPUT c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmro12.tfm INPUT ./dokumentation.gls INPUT dokumentation.gls INPUT ./dokumentation.gls @@ -1914,11 +1915,18 @@ INPUT dokumentation.run.xml OUTPUT dokumentation.run.xml INPUT c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-ec.enc INPUT c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc +INPUT c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc +INPUT c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc +INPUT c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmbx12.pfb +INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmex10.pfb +INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmmi12.pfb +INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmmi8.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr10.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr12.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr17.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr6.pfb +INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmr8.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmri12.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmri8.pfb INPUT c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmro12.pfb diff --git a/dokumentation.log b/dokumentation.log index bef6c71..38214fc 100644 --- a/dokumentation.log +++ b/dokumentation.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.9.29) 2 MAY 2023 12:08 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.9.29) 2 MAY 2023 13:12 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -1399,28 +1399,28 @@ Package microtype Info: Loading generic protrusion settings for font family (microtype) See the microtype manual for details. Package tocbasic Info: character protrusion at toc deactivated on input line 58. (./dokumentation.toc -LaTeX Font Info: Trying to load font information for OT1+lmr on input line 8. +LaTeX Font Info: Trying to load font information for OT1+lmr on input line 7. (c:/texlive/2022/texmf-dist/tex/latex/lm/ot1lmr.fd File: ot1lmr.fd 2015/05/01 v1.6.1 Font defs for Latin Modern ) -LaTeX Font Info: Trying to load font information for OML+lmm on input line 8. +LaTeX Font Info: Trying to load font information for OML+lmm on input line 7. (c:/texlive/2022/texmf-dist/tex/latex/lm/omllmm.fd File: omllmm.fd 2015/05/01 v1.6.1 Font defs for Latin Modern ) -LaTeX Font Info: Trying to load font information for OMS+lmsy on input line 8. +LaTeX Font Info: Trying to load font information for OMS+lmsy on input line 7. (c:/texlive/2022/texmf-dist/tex/latex/lm/omslmsy.fd File: omslmsy.fd 2015/05/01 v1.6.1 Font defs for Latin Modern ) -LaTeX Font Info: Trying to load font information for OMX+lmex on input line 8. +LaTeX Font Info: Trying to load font information for OMX+lmex on input line 7. (c:/texlive/2022/texmf-dist/tex/latex/lm/omxlmex.fd File: omxlmex.fd 2015/05/01 v1.6.1 Font defs for Latin Modern ) LaTeX Font Info: External font `lmex10' loaded for size -(Font) <12> on input line 8. +(Font) <12> on input line 7. LaTeX Font Info: External font `lmex10' loaded for size -(Font) <8> on input line 8. +(Font) <8> on input line 7. LaTeX Font Info: External font `lmex10' loaded for size -(Font) <6> on input line 8. +(Font) <6> on input line 7. ) \tf@toc=\write7 \openout7 = `dokumentation.toc'. @@ -1490,15 +1490,34 @@ chapter without number ] (./content/01kapitel.tex chapter 1. -) (./content/02kapitel.tex + pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored \relax -l.3 \chapter - {Beispiel Code-schnipsel einbinden} [1 +l.17 + [1 ] + +Package nag Warning: Environment eqnarray is obsolete. +(nag) Use amsmath's align instead on input line 18. + + +Package nag Warning: Environment eqnarray is obsolete. +(nag) Use amsmath's align instead on input line 27. + + +Package nag Warning: Environment eqnarray is obsolete. +(nag) Use amsmath's align instead on input line 36. + + +Overfull \vbox (7.89444pt too high) has occurred while \output is active [] + + [2]) (./content/02kapitel.tex +Overfull \vbox (7.89444pt too high) has occurred while \output is active [] + + [3] chapter 2. 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(microtype) See the microtype manual for details. - [2 + [4 ] Package acronym Info: Label `acro:AGPL' newly defined as it shall be overridden @@ -1517,51 +1536,45 @@ although it is yet undefined on input line 43. ) Overfull \vbox (7.89444pt too high) has occurred while \output is active [] - [3] + [5] LaTeX Warning: Empty bibliography on input line 100. -(./dokumentation.gls) [4 +(./dokumentation.gls) [6 ] (./ads/appendix.tex chapter without number -[5 +[7 ]) -No complaints by nag. -Overfull \vbox (7.89444pt too high) has occurred while \output is active [] - - [6] (./dokumentation.aux) +Package nag Warning: 3 complaints in total. -LaTeX Warning: There were undefined references. +Overfull \vbox (7.89444pt too high) has occurred while \output is active [] -Package biblatex Warning: Please (re)run Biber on the file: -(biblatex) dokumentation -(biblatex) and rerun LaTeX afterwards. - + [8] (./dokumentation.aux) Package logreq Info: Writing requests to 'dokumentation.run.xml'. \openout1 = `dokumentation.run.xml'. ) Here is how much of TeX's memory you used: - 45943 strings out of 475075 - 879798 string characters out of 5783001 - 2012701 words of memory out of 5000000 - 66294 multiletter control sequences out of 15000+600000 - 539194 words of font info for 106 fonts, out of 8000000 for 9000 + 46010 strings out of 475075 + 880855 string characters out of 5783001 + 2011702 words of memory out of 5000000 + 66310 multiletter control sequences out of 15000+600000 + 543786 words of font info for 127 fonts, out of 8000000 for 9000 1143 hyphenation exceptions out of 8191 - 113i,10n,109p,10708b,2457s stack positions out of 10000i,1000n,20000p,200000b,200000s -{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-ec.enc}{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc} -Output written on dokumentation.pdf (12 pages, 288736 bytes). + 113i,11n,109p,10708b,2457s stack positions out of 10000i,1000n,20000p,200000b,200000s +{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-ec.enc}{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc}{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc}{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc}{c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc} +Output written on dokumentation.pdf (14 pages, 337833 bytes). PDF statistics: - 216 PDF objects out of 1000 (max. 8388607) - 176 compressed objects within 2 object streams - 48 named destinations out of 1000 (max. 500000) - 42082 words of extra memory for PDF output out of 42996 (max. 10000000) + 275 PDF objects out of 1000 (max. 8388607) + 223 compressed objects within 3 object streams + 59 named destinations out of 1000 (max. 500000) + 44154 words of extra memory for PDF output out of 51595 (max. 10000000) diff --git a/dokumentation.lol b/dokumentation.lol index 012accd..9c05487 100644 --- a/dokumentation.lol +++ b/dokumentation.lol @@ -2,6 +2,6 @@ \addvspace {10\p@ } \addvspace {10\p@ } \addvspace {10\p@ } -\contentsline {lstlisting}{\numberline {2.1}Code-Beispiel}{2}{lstlisting.2.1}% -\contentsline {lstlisting}{\numberline {2.2}Python-Code}{2}{lstlisting.2.2}% +\contentsline {lstlisting}{\numberline {2.1}Code-Beispiel}{4}{lstlisting.2.1}% +\contentsline {lstlisting}{\numberline {2.2}Python-Code}{4}{lstlisting.2.2}% \addvspace {10\p@ } diff --git a/dokumentation.run.xml b/dokumentation.run.xml index 82e0d37..194bdbe 100644 --- a/dokumentation.run.xml +++ b/dokumentation.run.xml @@ -41,7 +41,7 @@ > ]> - + latex dokumentation.bcf @@ -63,7 +63,7 @@ english.lbx - + biber biber diff --git a/dokumentation.toc b/dokumentation.toc index 03f2252..1f8ebba 100644 --- a/dokumentation.toc +++ b/dokumentation.toc @@ -3,10 +3,16 @@ \contentsline {chapter}{\nonumberline Abbildungsverzeichnis}{III}{chapter*.3}% \contentsline {chapter}{\nonumberline Tabellenverzeichnis}{IV}{chapter*.4}% \contentsline {chapter}{\nonumberline Listings}{V}{chapter*.5}% -\contentsline {chapter}{\numberline {1}Das erste Kapitel}{1}{chapter.1}% -\contentsline {chapter}{\numberline {2}Beispiel Code-schnipsel einbinden}{2}{chapter.2}% -\contentsline {section}{\numberline {2.1}lorem ipsum}{2}{section.2.1}% -\contentsline {section}{\numberline {2.2}Verweis auf Code}{3}{section.2.2}% -\contentsline {chapter}{Glossar}{4}{section*.7}% -\contentsline {chapter}{\nonumberline Anhang}{5}{chapter*.9}% +\contentsline {chapter}{\numberline {1}Einleitung}{1}{chapter.1}% +\contentsline {section}{\numberline {1.1}Mathematische Formulierung}{2}{section.1.1}% +\contentsline {section}{\numberline {1.2}Komplexität NP-Schwer/Vollständig}{2}{section.1.2}% +\contentsline {section}{\numberline {1.3}verschiedene Arten des Rucksackproblems}{2}{section.1.3}% +\contentsline {subsection}{\numberline {1.3.1}Bounded Rucksackproblem}{3}{subsection.1.3.1}% +\contentsline {subsection}{\numberline {1.3.2}Unbounded Rucksackproblem}{3}{subsection.1.3.2}% +\contentsline {subsection}{\numberline {1.3.3}Multiples Rucksackproblem}{3}{subsection.1.3.3}% +\contentsline {chapter}{\numberline {2}Beispiel Code-schnipsel einbinden}{4}{chapter.2}% +\contentsline {section}{\numberline {2.1}lorem ipsum}{4}{section.2.1}% +\contentsline {section}{\numberline {2.2}Verweis auf Code}{5}{section.2.2}% +\contentsline {chapter}{Glossar}{6}{section*.7}% +\contentsline {chapter}{\nonumberline Anhang}{7}{chapter*.9}% \providecommand \tocbasic@end@toc@file {}\tocbasic@end@toc@file From cfa420a99a3808e1c0b526c7a0ba121a99afaf7b Mon Sep 17 00:00:00 2001 From: Timo Date: Tue, 2 May 2023 17:00:19 +0200 Subject: [PATCH 4/4] Chapter 1 Einleitung finished --- ads/acronyms.tex | 1 + ads/header.tex | 4 +- content/01kapitel.tex | 71 ++++++------ dokumentation-blx.bib | 11 ++ dokumentation.aux | 91 +++++++++++---- dokumentation.bbl | 230 +++++++++++++++++++++++++++++++++++++- dokumentation.bcf | 1 + dokumentation.blg | 74 ++++++++++-- dokumentation.fdb_latexmk | 40 ++++--- dokumentation.fls | 18 ++- dokumentation.lof | 1 + dokumentation.log | 85 +++++++++----- dokumentation.lol | 5 +- dokumentation.lot | 1 + dokumentation.run.xml | 17 ++- dokumentation.toc | 15 +-- sources/sources.bib | 106 ------------------ 17 files changed, 538 insertions(+), 233 deletions(-) create mode 100644 dokumentation-blx.bib delete mode 100644 sources/sources.bib diff --git a/ads/acronyms.tex b/ads/acronyms.tex index dff4cf1..8447bbc 100644 --- a/ads/acronyms.tex +++ b/ads/acronyms.tex @@ -22,4 +22,5 @@ \acro{API}{Application Programming Interface} \acro{WYSIWYG}{What You See Is What You Get} \acro{HTML}{HyperText Markup Language} +\acro{SSB}{Subset-Sum Problem} \end{acronym} diff --git a/ads/header.tex b/ads/header.tex index 843d44a..274de93 100644 --- a/ads/header.tex +++ b/ads/header.tex @@ -123,7 +123,7 @@ % Literaturverweise (sowohl deutsch als auch englisch) \iflang{de}{% \usepackage[ - backend=biber, % empfohlen. Falls biber Probleme macht: bibtex + backend=bibtex, % empfohlen. Falls biber Probleme macht: bibtex bibwarn=true, bibencoding=utf8, % wenn .bib in utf8, sonst ascii sortlocale=de_DE, @@ -132,7 +132,7 @@ } \iflang{en}{% \usepackage[ - backend=biber, % empfohlen. Falls biber Probleme macht: bibtex + backend=bibtex, % empfohlen. Falls biber Probleme macht: bibtex bibwarn=true, bibencoding=utf8, % wenn .bib in utf8, sonst ascii sortlocale=en_US, diff --git a/content/01kapitel.tex b/content/01kapitel.tex index 6f0bfe9..fbb7648 100644 --- a/content/01kapitel.tex +++ b/content/01kapitel.tex @@ -1,53 +1,60 @@ %!TEX root = ../dokumentation.tex \chapter{Einleitung} -Bei dem Rucksackproblem handelt es sich um eine einfach zu beschreibende, aber auf keinen Fall einfach zu lösende Aufgabe der diskreten Mathematik. Zur Lösung des Problems werden meist heuristische Verfahren oder Methoden der dynamischen Optimierung verwendet. (Vgl. Luderer, S.104) -Anschaulich geht es bei dem Rucksackproblem um einen Rucksack, der höchstens ein bestimmtes vorgegebenes Gewicht $T$ tragen kann. $T$ wird als Gewichtsschranke bezeichnet. Neben dem Rucksack gibt es eine Menge von Gegenschänden. Diese werden als n Objekten bezeichnet. Jedes Objekt hat ein Gewicht und einen Profit. Untersucht wird nun die Teilmengen der Objekte, die die Gewichtsschranke $T$ nicht verletzten, dabei aber einen möglichst großen Gesamtprofit haben. Das heißt, die Summe der Profite der eingepackten Objekte maximal wird. -Rein intuitiv erscheint es sinnvoll, die Objekte mit dem größten Profit pro Gewichtseinheit zuerst auszuwählen. Das Verhältnis zwischen Profit und Gewicht eines Objektes wird auch als Profitdichte bezeichnet. Die intuitive Lösung ist allerdings nicht immer die beste, da die Gewichtsschranke $T$ teilweise stark unterschritten wird, ein weiteres Objekt jedoch die Gewichtsschranke überschreiten würde. Das heißt, es wird möglicherweise Platz verschwendet. -Die beste Lösung kann garantiert durch Ausprobieren aller möglichen Kombinationen gefunden werden. Allerdings gibt es sehr viele Kombinationsmöglichkeiten. Für jedes einzelne Objekt ist zu entscheiden, ob es in den Rucksack gepackt wird oder nicht. Somit gibt es pro Objekt zwei Möglichkeiten. Bei $n$ Objekten ergeben sich $2^n$ Kombinationsmöglichkeiten. Durch das exponentielle Wachstum der Kombinationsmöglichkeiten „explodiert“ die Anzahl der möglichen Teilmengen. Beispielsweise wäre das Rucksackproblem bei 60 Objekten durch diesen Ansatz mit heutigen Mitteln kaum lösbar, da es bereits $2^60 = 1 152 921 504 606 846 976$ Kombinationsmöglichkeiten gibt. Angenommen, ein Computer schafft es, eine Milliarde Teilmengen pro Sekunden zu berechnen und zu testen. Dann benötigte dieser trotzdem noch über 36 Jahre, um alle Teilmengen auszuprobieren. (vgl. Beier, S.405ff) - +Bei dem Rucksackproblem handelt es sich um eine einfach zu beschreibende, aber auf keinen Fall einfach zu lösende Aufgabe der diskreten Mathematik. Zur Lösung des Problems werden meist heuristische Verfahren oder Methoden der dynamischen Optimierung verwendet. \cite[vgl.][]{Luderer2017} \\ +Anschaulich geht es bei dem Rucksackproblem um einen Rucksack, der höchstens ein bestimmtes vorgegebenes Gewicht $T$ tragen kann. $T$ wird als Gewichtsschranke bezeichnet. Neben dem Rucksack gibt es eine Menge von Gegenschänden. Diese werden als $n$ Objekten bezeichnet. Jedes Objekt hat ein Gewicht und einen Profit. Untersucht wird nun die Teilmengen der Objekte, die die Gewichtsschranke $T$ nicht verletzten, dabei aber einen möglichst großen Gesamtprofit haben. Das heißt, die Summe der Profite der eingepackten Objekte maximal wird. \\ +Rein intuitiv erscheint es sinnvoll, die Objekte mit dem größten Profit pro Gewichtseinheit zuerst auszuwählen. Das Verhältnis zwischen Profit und Gewicht eines Objektes wird auch als Profitdichte bezeichnet. Die intuitive Lösung ist allerdings nicht immer die beste, da die Gewichtsschranke $T$ teilweise stark unterschritten wird, ein weiteres Objekt jedoch die Gewichtsschranke überschreiten würde. Das heißt, es wird möglicherweise Platz verschwendet.\\ +Die beste Lösung kann garantiert durch Ausprobieren aller möglichen Kombinationen gefunden werden. Allerdings gibt es sehr viele Kombinationsmöglichkeiten. Für jedes einzelne Objekt ist zu entscheiden, ob es in den Rucksack gepackt wird oder nicht. Somit gibt es pro Objekt zwei Möglichkeiten. Bei $n$ Objekten ergeben sich $2^n$ Kombinationsmöglichkeiten. Durch das exponentielle Wachstum der Kombinationsmöglichkeiten „explodiert“ die Anzahl der möglichen Teilmengen. Beispielsweise wäre das Rucksackproblem bei 60 Objekten durch diesen Ansatz mit heutigen Mitteln kaum lösbar, da es bereits $2^{60} = 1 152 921 504 606 846 976$ Kombinationsmöglichkeiten gibt. Angenommen, ein Computer schafft es, eine Milliarde Teilmengen pro Sekunden zu berechnen und zu testen. Dann benötigte dieser trotzdem noch über 36 Jahre, um alle Teilmengen auszuprobieren. \cite[vgl.][]{Vocking2008} \\ In dieser Arbeit wird auf das klassische binäre Rucksackproblem, auch 0-1 Rucksackproblem genannt, eingegangen. Dabei gibt es nur die Möglichkeit ein Objekt einzupacken oder nicht. Kopien eines Objektes sind dabei nicht erlaubt. Varianten des Rucksackproblems, in denen Kopien eines Objektes erlaubt sind, wie zum Beispiel das Bounded Rucksackproblem, werden in wenigen Sätzen erläutert. \section{Mathematische Formulierung} -(Vgl. Matello, S.214) Mathematisch kann das Rucksackproblem durch ein Ganzzahliges Lineares Programm abgebildete werden. Dabei gebe es ein Rucksack oder anderes Gefäß mit Kapazität $c$ und $n$ Objekt beziehungsweise Gegenstände mit - Profit $p_j,\ \ \ j=1,\ldots,n$ und - Gewicht $w_j,\ \ j=1,\ldots,n$. -Die einfachste Form des Rucksackproblems, das 0-1 Rucksackproblem, kann mathematisch folgendermaßen dargestellt werden: - -\begin{eqnarray} - Maximize - z=\sum_{j=1}^{n}{p_jx_j} - Subject to - \sum_{j=1}^{n}{w_jx_j\le c} - x_j\in\{0,1\},\ \ j=1,\ldots,n -\end{eqnarray} +\begin{itemize} + \item Profit $p_j$ mit $j=1,\ldots,n$ und + \item Gewicht $w_j$ mit $j=1,\ldots,n$. +\end{itemize} +Die einfachste Form des Rucksackproblems, das 0-1 Rucksackproblem, kann mathematisch folgendermaßen dargestellt werden: \cite[vgl.][]{Martello1987} + +Maximize +\begin{equation} +z=\sum_{j=1}^{n}{p_jx_j} +\end{equation} +Subject to +\begin{equation} \label{eqn:subjectTo} +\sum_{j=1}^{n}{w_jx_j\le c} +\end{equation} +\begin{equation} \label{eqn:bedingung} +x_j\in\{0,1\},\ \ j=1,\ldots,n +\end{equation} -Dabei ist $x_j = 1$, wenn das Objekt an der Stelle $j$ ausgewählt wird, wird das Objekt nicht ausgewählt ist $x_j = 0$. Da Profite und Gewichte positiv sind, wird ohne Beschränkung der Allgemeinheit angenommen, dass gilt: +Dabei ist $x_j = 1$, wenn das Objekt an der Stelle $j$ ausgewählt wird. Wird das Objekt nicht ausgewählt ist $x_j = 0$. Da Profite und Gewichte positiv sind, wird ohne Beschränkung der Allgemeinheit angenommen, dass gilt: \cite[vgl.][]{Martello1987} \begin{eqnarray} - \sum_{j=1}^{n}{w_j>c} - w_j\le c,\ \ j=1,\ldots,n +\sum_{j=1}^{n}{w_j>c} \\ +w_j\le c,\ \ j=1,\ldots,n \end{eqnarray} \section{Komplexität NP-Schwer/Vollständig} -(Vgl. Martello, S.233) -Das Rucksackproblem ist NP-Schwer und NP-Vollständig, weil bewiesen wurde, dass das Teilsummenproblem (Subset-Sum Problem SSB) ebenfalls NP-Schwer und NP-Vollständig ist. Beweis aufführen? vgl. Martello S.233f Das Teilsummenproblem ein Spezialfall des Rucksackproblems unter der Bedingung, dass $p_j=w_j$ für alle $j$. Das Teilsummenproblem kann mathematisch wie folgt dargestellt werden: -\begin{eqnarray} -maximize -z=\sum_{j=1}^{n}{w_jx_j} +Das Rucksackproblem ist NP-Schwer und NP-Vollständig, weil bewiesen wurde, dass das Teilsummenproblem (englisch: \ac{SSB}) ebenfalls NP-Schwer und NP-Vollständig ist. \textcolor{red}{Beweis aufführen? vgl. Martello S.233f} Das Teilsummenproblem ein Spezialfall des Rucksackproblems unter der Bedingung, dass $p_j=w_j$ für alle $j \in \{1, \dots n\}$. Das Teilsummenproblem kann mathematisch wie folgt dargestellt werden: \cite[vgl.][]{Martello1987} +\newpage +Maximize +\begin{equation} +z=\sum_{j=1}^{n}{w_jx_j} +\end{equation} Subject to -\sum_{j=1}^{n}{w_jx_j\le c} -x_j\in0,1,\ \ j=1,\ldots,n +\begin{eqnarray} +\sum_{j=1}^{n}{w_jx_j\le c}\\ +x_j\in\{0,1\},\ \ j=1,\ldots,n \end{eqnarray} -Dadurch kann das Rucksackproblem wiederum auf das Teilsummenproblem reduziert werden. -\section{verschiedene Arten des Rucksackproblems} +\textcolor{red}{stimmt der satz so?:} Dadurch kann das Rucksackproblem wiederum auf das Teilsummenproblem reduziert werden. + +\section{Verschiedene Arten des Rucksackproblems} Neben dem klassischen 0-1 Rucksackproblem gibt es weitere Formen des Problems, die im Folgenden in einigen Sätzen beschrieben werden. Es werden nicht alle Varianten des Rucksackproblems aufgeführt. \subsection{Bounded Rucksackproblem} -Beim Bounded Rucksackproblemen können endlich viele identische Kopien $b_j$ eines Objektes $j$ eingepackt werden. Dadurch ändert sich lediglich die Bedingung der Formel Referenz einfügen von $x_j\in\{0,1\}$ zu $0\le\ x_j\le\ b_j$. Dadurch kann $x_j$ jeden ganzzahligen Wert $0$ und $b_j$ annehmen. Vgl. Martello1990, S.81f Das Unbounded Rucksackproblem kann in das klassische 0-1 Rucksackproblem transformiert werden vgl. Schury, S.17. +Beim Bounded Rucksackproblem können endlich viele identische Kopien $b_j$ eines Objektes $j$ eingepackt werden. Dadurch ändert sich lediglich die Bedingung \ref{eqn:bedingung} der Formel \ref{eqn:subjectTo} von $x_j\in\{0,1\}$ zu $0\le\ x_j\le\ b_j$. Dadurch kann $x_j$ jeden ganzzahligen Wert $0$ und $b_j$ annehmen. \cite[vgl.][]{Martello1990} Das Unbounded Rucksackproblem kann in das klassische 0-1 Rucksackproblem transformiert werden \cite[vgl.][]{Schury2013}. \subsection{Unbounded Rucksackproblem} -Beim Unbounded Rucksackproblem können im Gegensatz zum Bounded Rucksackproblem unendlich viele identische Kopien $b_j$ eines Objektes $j$ eingepackt werden. Dadurch ändert sich die Bedingung der Formel Referenz einfügen von $x_j\in\{0,1\}$ zu $x_j\geq\ 0$. Dadurch kann $x_j$ jeden ganzzahligen Wert größer als $0$ annehmen. Das Unbounded Rucksackproblem kann in das 0-1 Rucksackproblem transformiert werden. vgl. Schury, S.17 +Beim Unbounded Rucksackproblem können im Gegensatz zum Bounded Rucksackproblem unendlich viele identische Kopien $b_j$ eines Objektes $j$ eingepackt werden. Dadurch ändert sich die Bedingung \ref{eqn:bedingung} der Formel \ref{eqn:subjectTo} von $x_j\in\{0,1\}$ zu $x_j\geq\ 0$. Dadurch kann $x_j$ jeden ganzzahligen Wert größer als $0$ annehmen. Das Unbounded Rucksackproblem kann in das 0-1 Rucksackproblem transformiert werden. \cite[vgl.][]{Schury2013} \subsection{Multiples Rucksackproblem} -Bei dem Multiplen Rucksackproblem stehen anstatt eines einzelnen Rucksacks oder Gefäß $m$ Rucksäcke mit unterschiedlichen Kapazitäten $c_m$ zur Verfügung. Alle Gegenstände sollen nun in die Rucksäcke verteilt werden, sodass keine Gewichtsschranke überschritten und der Profit maximal wird. Dabei wird aus der Binärvariable $x_j$ des 0-1 Rucksackproblems eine Binärvariablenmatrix. Diese beschreibt, welches Objekt $j$ in welchen Rucksack $i$ eingepackt wird. (vgl. Martello1990, S.157ff) +Bei dem Multiplen Rucksackproblem stehen anstatt eines einzelnen Rucksacks oder Gefäß $m$ Rucksäcke mit unterschiedlichen Kapazitäten $c_m$ zur Verfügung. Alle Gegenstände sollen nun in die Rucksäcke verteilt werden, sodass keine Gewichtsschranke überschritten und der Profit maximal wird. Dabei wird aus der Binärvariable $x_j$ des 0-1 Rucksackproblems eine Binärvariablenmatrix. Diese beschreibt, welches Objekt $j$ in welchen Rucksack $i$ eingepackt wird. \cite[vgl.][]{Martello1990} diff --git a/dokumentation-blx.bib b/dokumentation-blx.bib new file mode 100644 index 0000000..8b7e3e0 --- /dev/null +++ b/dokumentation-blx.bib @@ -0,0 +1,11 @@ +@Comment{$ biblatex control file $} +@Comment{$ biblatex bcf format version 3.9 $} +% Do not modify this file! +% +% This is an auxiliary file used by the 'biblatex' package. +% This file may safely be deleted. It will be recreated as +% required. + +@Control{biblatex-control, + options = {3.9:0:0:1:0:1:1:0:0:0:0:1:3:1:3:1:0:0:3:1:79:+:+:nty}, +} diff --git a/dokumentation.aux b/dokumentation.aux index 9039e7f..a3b7d5c 100644 --- a/dokumentation.aux +++ b/dokumentation.aux @@ -19,6 +19,9 @@ \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} \providecommand\BKM@entry[2]{} +\bibstyle{biblatex} +\bibdata{dokumentation-blx,bibliographie} +\citation{biblatex-control} \abx@aux@refcontext{nty/global//global/global} \gdef \LT@i {\LT@entry {1}{253.3125pt}\LT@entry @@ -44,6 +47,7 @@ \newacro{API}[\AC@hyperlink{API}{API}]{Application Programming Interface} \newacro{WYSIWYG}[\AC@hyperlink{WYSIWYG}{WYSIWYG}]{What You See Is What You Get} \newacro{HTML}[\AC@hyperlink{HTML}{HTML}]{HyperText Markup Language} +\newacro{SSB}[\AC@hyperlink{SSB}{SSB}]{Subset-Sum Problem} 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It will be recreated by -% biber as required. +% This file may safely be deleted. It will be recreated as +% required. % \begingroup \makeatletter @@ -16,5 +16,229 @@ {} \endgroup -\endinput +\datalist[entry]{nty/global//global/global} + \entry{Luderer2017}{book}{} + \name{author}{1}{}{% + {{hash=LB}{% + family={Luderer}, + familyi={L\bibinitperiod}, + given={B.}, + giveni={B\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {Springer Fachmedien Wiesbaden}% + } + \strng{namehash}{LB1} + \strng{fullhash}{LB1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{sortinit}{L} + \field{sortinithash}{L} + \field{isbn}{9783658191870} + \field{pages}{104\bibrangedash 105} + \field{title}{Facetten der Wirtschaftsmathematik: Eine unterhaltsame + Einf{\"u}hrung ganz ohne Formeln} + \verb{url} + \verb https://books.google.de/books?id=F256swEACAAJ + \endverb + \field{year}{2017} + \endentry + + \entry{Martello1987}{incollection}{} + \name{author}{2}{}{% + {{hash=MS}{% + family={Martello}, + familyi={M\bibinitperiod}, + given={Silvano}, + giveni={S\bibinitperiod}, + }}% + {{hash=TP}{% + family={Toth}, + familyi={T\bibinitperiod}, + given={Paolo}, + giveni={P\bibinitperiod}, + }}% + } + \name{editor}{4}{}{% + {{hash=MS}{% + family={Martello}, + familyi={M\bibinitperiod}, + given={Silvano}, + giveni={S\bibinitperiod}, + }}% + {{hash=LG}{% + family={Laporte}, + familyi={L\bibinitperiod}, + given={Gilbert}, + giveni={G\bibinitperiod}, + }}% + {{hash=MM}{% + family={Minoux}, + familyi={M\bibinitperiod}, + given={Michel}, + giveni={M\bibinitperiod}, + }}% + {{hash=RC}{% + family={Ribeiro}, + familyi={R\bibinitperiod}, + given={Celso}, + giveni={C\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {North-Holland}% + } + \strng{namehash}{MSTP1} + \strng{fullhash}{MSTP1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{sortinit}{M} + \field{sortinithash}{M} + \field{abstract}{% + Publisher Summary This chapter presents a great variety of practical + problems that are represented by a set of entities, each having an associated + value, from which one or more subsets has to be selected in such a way that + the sum of the values of the selected entities is maximized and some + predefined conditions. The most common condition is obtained by associating a + size to each entity and establishing that the sum of the entity sizes in each + subset does not exceed some prefixed bound. These problems are generally + called knapsack problems, as they recall the situation of a hitch-hiker + having to fill up knapsack by selecting from among various possible objects + those that provides the maximum comfort. The entities are called as items and + their number is indicated by n. The value and size associated with the j-th + item are called as profit and weight. The majority of problems are considered + as single knapsack problems, where one container must be filled with an + optimal subset of items. The chapter reviews both exact and approximate + algorithms for the solution of the most important types of knapsack problems. + Exact solutions are obtained through branch-and-bound algorithms or dynamic + programming, or a combination of the two approaches. The performances of + these algorithms are evaluated by computational experiments with randomly + generated test problems. Approximate algorithms are of the greedy type or are + based on scaling. They are evaluated on the basis of their worst-case + performance.% + } + \field{booktitle}{Surveys in Combinatorial Optimization} + \verb{doi} + \verb https://doi.org/10.1016/S0304-0208(08)73237-7 + \endverb + \field{issn}{0304-0208} + \field{pages}{213\bibrangedash 257} + \field{series}{North-Holland Mathematics Studies} + \field{title}{Algorithms for Knapsack Problems} + \verb{url} + \verb https://www.sciencedirect.com/science/article/pii/S0304020808732377 + \endverb + \field{volume}{132} + \field{year}{1987} + \endentry + + \entry{Martello1990}{article}{} + \name{author}{2}{}{% + {{hash=MS}{% + family={Martello}, + familyi={M\bibinitperiod}, + given={Silvano}, + giveni={S\bibinitperiod}, + }}% + {{hash=TP}{% + family={Toth}, + familyi={T\bibinitperiod}, + given={Paolo}, + giveni={P\bibinitperiod}, + }}% + } + \keyw{integer programming, knapsack problem, branch-and-bound} + \strng{namehash}{MSTP1} + \strng{fullhash}{MSTP1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{sortinit}{M} + \field{sortinithash}{M} + \field{abstract}{% + Given n types, each having an associated profit and weight, and a container + of given capacity, the unbounded knapsack problem is to determine the number + of items of each type to be selected so that the corresponding total profit + is a maximum and the corresponding total weight does not exceed the capacity. + We present upper bounds, dominance relations, and an approach—based on the + definition of a core problem—to the exact solution of very large instances + of the problem. We give the results of computational experiments on randomly + generated test problems involving up to 250 000 item types.% + } + \verb{doi} + \verb https://doi.org/10.1016/0167-6377(90)90035-4 + \endverb + \field{issn}{0167-6377} + \field{number}{1} + \field{pages}{15\bibrangedash 20} + \field{title}{An exact algorithm for large unbounded knapsack problems} + \verb{url} + \verb https://www.sciencedirect.com/science/article/pii/0167637790900354 + \endverb + \field{volume}{9} + \field{journaltitle}{Operations Research Letters} + \field{year}{1990} + \endentry + \entry{Schury2013}{thesis}{} + \name{author}{1}{}{% + {{hash=SC}{% + family={Schury}, + familyi={S\bibinitperiod}, + given={Christoph}, + giveni={C\bibinitperiod}, + }}% + } + \list{language}{1}{% + {German}% + } + \keyw{Rucksackproblem, Kollektiventscheidung, Online-Publikation} + \strng{namehash}{SC1} + \strng{fullhash}{SC1} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{sortinit}{S} + \field{sortinithash}{S} + \field{pages}{IV, 55} + \field{title}{Social Choice Methoden zur Lösung von mehrdimensionalen + Rucksackproblemen / Social Choice Methods for the solution of the + multimensional knapsack problem} + \list{institution}{1}{% + {Universität Graz}% + } + \field{type}{mathesis} + \field{year}{2013} + \endentry + + \entry{Vocking2008}{book}{} + \name{editor}{2}{}{% + {{hash=VB}{% + family={Vöcking}, + familyi={V\bibinitperiod}, + given={B.}, + giveni={B\bibinitperiod}, + }}% + {{hash=w}{% + family={weitere}, + familyi={w\bibinitperiod}, + }}% + } + \list{publisher}{1}{% + {Springer}% + } + \strng{namehash}{VBw1} + \strng{fullhash}{VBw1} + \field{labelnamesource}{editor} + \field{labeltitlesource}{title} + \field{sortinit}{V} + \field{sortinithash}{V} + \field{isbn}{9783540763932} + \field{pages}{405\bibrangedash 413} + \field{title}{Taschenbuch der Algorithmen} + \list{location}{1}{% + {Berlin, Heidelberg}% + } + \field{year}{2008} + \endentry +\enddatalist +\endinput diff --git a/dokumentation.bcf b/dokumentation.bcf index b52a972..2ad4735 100644 --- a/dokumentation.bcf +++ b/dokumentation.bcf @@ -2352,6 +2352,7 @@ bibliographie.bib + Luderer2017 diff --git a/dokumentation.blg b/dokumentation.blg index f8e7cf0..e86f332 100644 --- a/dokumentation.blg +++ b/dokumentation.blg @@ -1,9 +1,65 @@ -[4] Config.pm:306> INFO - This is Biber 2.18 -[5] Config.pm:309> INFO - Logfile is 'dokumentation.blg' -[1605] biber-MSWIN32:340> INFO - === -[2117] Biber.pm:418> INFO - Reading 'dokumentation.bcf' -[3780] Biber.pm:978> INFO - Found 0 citekeys in bib section 0 -[4007] bbl.pm:654> INFO - Writing 'dokumentation.bbl' with encoding 'UTF-8' -[4009] bbl.pm:757> INFO - Output to dokumentation.bbl -[4013] Biber.pm:130> WARN - The file 'dokumentation.bcf' does not contain any citations! -[4014] Biber.pm:132> INFO - WARNINGS: 1 +This is BibTeX, Version 0.99d (TeX Live 2022) +Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 +The top-level auxiliary file: dokumentation.aux +The style file: biblatex.bst +Reallocated glb_str_ptr (elt_size=4) to 20 items from 10. +Reallocated global_strs (elt_size=200001) to 20 items from 10. +Reallocated glb_str_end (elt_size=4) to 20 items from 10. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated wiz_functions (elt_size=4) to 6000 items from 3000. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Database file #1: dokumentation-blx.bib +Database file #2: bibliographie.bib +Biblatex version: 3.18 +Reallocated wiz_functions (elt_size=4) to 9000 items from 6000. +Reallocated singl_function (elt_size=4) to 100 items from 50. +You've used 6 entries, + 6399 wiz_defined-function locations, + 1256 strings with 12391 characters, +and the built_in function-call counts, 38615 in all, are: += -- 2953 +> -- 669 +< -- 227 ++ -- 1138 +- -- 223 +* -- 2494 +:= -- 4257 +add.period$ -- 0 +call.type$ -- 6 +change.case$ -- 143 +chr.to.int$ -- 42 +cite$ -- 10 +duplicate$ -- 4263 +empty$ -- 3286 +format.name$ -- 337 +if$ -- 7920 +int.to.chr$ -- 0 +int.to.str$ -- 19 +missing$ -- 0 +newline$ -- 214 +num.names$ -- 321 +pop$ -- 1804 +preamble$ -- 1 +purify$ -- 166 +quote$ -- 0 +skip$ -- 1229 +stack$ -- 0 +substring$ -- 4837 +swap$ -- 906 +text.length$ -- 163 +text.prefix$ -- 5 +top$ -- 1 +type$ -- 191 +warning$ -- 0 +while$ -- 581 +width$ -- 0 +write$ -- 209 diff --git a/dokumentation.fdb_latexmk b/dokumentation.fdb_latexmk index d0d09d3..90522bb 100644 --- a/dokumentation.fdb_latexmk +++ b/dokumentation.fdb_latexmk @@ -1,29 +1,34 @@ # Fdb version 3 -["biber dokumentation"] 1683025888 "dokumentation.bcf" "dokumentation.bbl" "dokumentation" 1683026027 - "dokumentation.bcf" 1683026022 107388 34346f1d15652dcd41bb57252a5d6744 "pdflatex" +["bibtex dokumentation"] 1683039528 "dokumentation.aux" "dokumentation.bbl" "dokumentation" 1683039538 + "bibliographie.bib" 1683039504 6254 f6e741645765d70b51df2b0c652bb58b "" + "c:/texlive/2022/texmf-dist/bibtex/bst/biblatex/biblatex.bst" 1664465198 65483 3b5a53ada692a06ce416d088de6a391b "" + "dokumentation-blx.bib" 1683039535 340 25674e554aa25e98ea59ec40797d70f1 "pdflatex" + "dokumentation.aux" 1683039535 11176 5f4293cc918c8eff5e1d2de287e8faca "pdflatex" (generated) "dokumentation.bbl" "dokumentation.blg" -["cusdep glo gls dokumentation"] 0 "dokumentation.glo" "dokumentation.gls" "dokumentation" 1683026027 +["cusdep glo gls dokumentation"] 0 "dokumentation.glo" "dokumentation.gls" "dokumentation" 1683039538 "dokumentation.glo" 0 -1 0 "pdflatex" (generated) "dokumentation.gls" -["pdflatex"] 1683025965 "d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/dokumentation.tex" "dokumentation.pdf" "dokumentation" 1683026027 - "ads/acronyms.tex" 1683020997 1196 22948ccac06f5845c6dbe531485c5ebb "" +["pdflatex"] 1683039529 "d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/dokumentation.tex" "dokumentation.pdf" "dokumentation" 1683039538 + "ads/acronyms.tex" 1683038396 1228 7c076764818761f4bed0e2d04c27abc9 "" "ads/appendix.tex" 1683020997 2300 d497f5c07b158c2a1bfdff0eb60ee0f4 "" "ads/deckblatt.tex" 1683021923 1219 48d2435b494a85b5b4f5ffacb2542253 "" "ads/einstellungen_liste.tex" 1683020997 543 68240c64f3bb48f50cbd7753cdbe47a8 "" "ads/glossary.tex" 1683020997 415 8fcc3eec5a4a302cc681b41d86b74292 "" - "ads/header.tex" 1683020997 6116 9198c1e251ab3f6b1a06bf6bb3664357 "" + "ads/header.tex" 1683037342 6118 970d60aa4cbecbd3d089aef8e104a5e9 "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-ec.enc" 1664476421 2375 baa924870cfb487815765f9094cf3728 "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc" 1664476421 3486 c7eadf5dcc57b3b2d11736679f6636ba "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc" 1664476421 2405 5dcf2c1b967ee25cc46c58cd52244aed "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc" 1664476421 2840 216e6e45ad352e2456e1149f28885bee "" "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc" 1664476421 2327 9d6df24f9c4f7368395224341a95523a "" + "c:/texlive/2022/texmf-dist/fonts/enc/dvips/lm/lm-ts1.enc" 1664476421 3031 6c4d3515bf7115d8518af1c9ab97ca44 "" "c:/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1664466751 3524 cb3e574dea2d1052e39280babc910dc8 "" "c:/texlive/2022/texmf-dist/fonts/tfm/jknappen/ec/ecrm1200.tfm" 1664466279 3584 f80ddd985bd00e29e9a6047ebd9d4781 "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1664465627 1288 655e228510b4c2a1abe905c368440826 "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmbx12.tfm" 1664476421 12088 d750ac78274fa7c9f73ba09914c04f8a "" + "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmcsc10.tfm" 1664476421 11276 116dd5bea6621ce4a1999f96d876084c "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmr10.tfm" 1664476421 12056 7e13df7fe4cbce21b072ba7c4f4deb6e "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmr12.tfm" 1664476421 12092 7b1546e2d096cfd5dcbd4049b0b1ec2e "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ec-lmr17.tfm" 1664476421 12156 ca1ae6a3c8564e89597f1f993fba1608 "" @@ -45,7 +50,9 @@ "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/rm-lmr12.tfm" 1664476421 11888 6841b91e46b65cf41a49b160e6e74130 "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/rm-lmr6.tfm" 1664476421 11836 e3b6ce3e601aec94f64a536e7f4224d5 "" "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/rm-lmr8.tfm" 1664476421 11864 309fd7f43e4a0ba39f6f7644d76e8edf "" + "c:/texlive/2022/texmf-dist/fonts/tfm/public/lm/ts1-lmr12.tfm" 1664476421 1596 1d4548788e389ded56d0b01b28377882 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmbx12.pfb" 1664476421 116908 9a7a12d1e4df280ea0ab4aad83cdef88 "" + "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmcsc10.pfb" 1664476421 116427 4909e80035dae5d5a4ec2b5ddd9649dd "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmex10.pfb" 1664476421 23055 2e5b42921de910eaa97b85df04ca4891 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmmi12.pfb" 1664476421 30696 3da7e46ecf5615eedebd9f27dab4d9d7 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmmi8.pfb" 1664476421 30635 833ec815d446ec453a4913fc26d24cbc "" @@ -60,6 +67,7 @@ "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmssbx10.pfb" 1664476421 119663 1a3a2206591ddc98c6d6c6271a282516 "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmsy10.pfb" 1664476421 27863 09ce3735688ffde955e72da27c95b61a "" "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmtt10.pfb" 1664476421 113227 d3d1adc024746ff57b20efba82c6d365 "" + "c:/texlive/2022/texmf-dist/fonts/type1/public/lm/lmtt12.pfb" 1664476421 110323 19372944fba298bb4c4758abff959582 "" "c:/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1664476938 71627 94eb9990bed73c364d7f53f960cc8c5b "" "c:/texlive/2022/texmf-dist/tex/generic/atbegshi/atbegshi.sty" 1664464965 24708 5584a51a7101caf7e6bbf1fc27d8f7b1 "" "c:/texlive/2022/texmf-dist/tex/generic/babel-english/english.ldf" 1664465018 7008 9ff5fdcc865b01beca2b0fe4a46231d4 "" @@ -164,6 +172,7 @@ "c:/texlive/2022/texmf-dist/tex/latex/biblatex/biblatex.cfg" 1664465198 69 249fa6df04d948e51b6d5c67bea30c42 "" "c:/texlive/2022/texmf-dist/tex/latex/biblatex/biblatex.def" 1664465198 92456 21e687f013958a6cb57adaa61a04572a "" "c:/texlive/2022/texmf-dist/tex/latex/biblatex/biblatex.sty" 1664465198 526811 a1f8c6dfa1788c26d4b7587a2e99a625 "" + "c:/texlive/2022/texmf-dist/tex/latex/biblatex/blx-bibtex.def" 1664465198 24613 9fabc69e3860a7fb0186ecefade563ce "" "c:/texlive/2022/texmf-dist/tex/latex/biblatex/blx-case-expl3.sty" 1664465198 8433 72f8188742e7214b7068f345cd0287ac "" "c:/texlive/2022/texmf-dist/tex/latex/biblatex/blx-compat.def" 1664465198 13919 5426dbe90e723f089052b4e908b56ef9 "" "c:/texlive/2022/texmf-dist/tex/latex/biblatex/blx-dm.def" 1664465198 32455 8d3e554836db11aab80a8e11be62e1b1 "" @@ -253,6 +262,7 @@ "c:/texlive/2022/texmf-dist/tex/latex/lm/t1lmr.fd" 1664476421 1867 996fe743d88a01aca041ed22cc10e1bb "" "c:/texlive/2022/texmf-dist/tex/latex/lm/t1lmss.fd" 1664476421 1641 f48673bce41ac279099d0c9523459753 "" "c:/texlive/2022/texmf-dist/tex/latex/lm/t1lmtt.fd" 1664476421 2682 555da1faa2e266801e4b221d01a42cb5 "" + "c:/texlive/2022/texmf-dist/tex/latex/lm/ts1lmr.fd" 1664476421 1914 884882d7ebb0fd65cea93fca77ff6f5a "" "c:/texlive/2022/texmf-dist/tex/latex/logreq/logreq.def" 1664476441 1620 fb1c32b818f2058eca187e5c41dfae77 "" "c:/texlive/2022/texmf-dist/tex/latex/logreq/logreq.sty" 1664476441 6187 b27afc771af565d3a9ff1ca7d16d0d46 "" "c:/texlive/2022/texmf-dist/tex/latex/mfirstuc/mfirstuc.sty" 1664476784 13639 7611f25667b1314d0db82582a6390613 "" @@ -297,25 +307,25 @@ "c:/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1664479675 4635069 7677ee69604aae14ed48b153c111a7f7 "" "c:/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1664479754 3058675 f649c66cce6a93d69f7b71a9c0c74319 "" "c:/texlive/2022/texmf.cnf" 1664479650 713 e69b156964470283e0530f5060668171 "" - "content/01kapitel.tex" 1683025841 6045 61e8239feaa6390abc455d3e604e8feb "" + "content/01kapitel.tex" 1683039525 6423 356111eb9f1e18539d43a36b5672b329 "" "content/02kapitel.tex" 1683020997 2961 0b3ff97ffcf1a769e39c4fc613c93d20 "" "d:/10_Daten/21_Studium(Daten)/Semester04/DiskreteMathematik-RucksackproblemLaTex/dokumentation.tex" 1683022073 2141 41b8bc6d0729e593fa006ed0b61ac698 "" - "dokumentation.aux" 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dokumentation.bcf + dokumentation.aux + dokumentation-blx.bib dokumentation.bbl @@ -52,6 +53,7 @@ blx-dm.def blx-compat.def + blx-bibtex.def biblatex.def standard.bbx numeric.bbx @@ -64,13 +66,14 @@ - biber + bibtex - biber + bibtex + dokumentation - dokumentation.bcf + dokumentation.aux dokumentation.bbl @@ -79,10 +82,14 @@ dokumentation.bbl - dokumentation.bcf + dokumentation.aux + dokumentation-blx.bib bibliographie.bib + + biblatex.bst + diff --git a/dokumentation.toc b/dokumentation.toc index 1f8ebba..b85f14e 100644 --- a/dokumentation.toc +++ b/dokumentation.toc @@ -6,13 +6,14 @@ \contentsline {chapter}{\numberline {1}Einleitung}{1}{chapter.1}% \contentsline {section}{\numberline {1.1}Mathematische Formulierung}{2}{section.1.1}% \contentsline {section}{\numberline {1.2}Komplexität NP-Schwer/Vollständig}{2}{section.1.2}% -\contentsline {section}{\numberline {1.3}verschiedene Arten des Rucksackproblems}{2}{section.1.3}% +\contentsline {section}{\numberline {1.3}Verschiedene Arten des Rucksackproblems}{3}{section.1.3}% \contentsline {subsection}{\numberline {1.3.1}Bounded Rucksackproblem}{3}{subsection.1.3.1}% \contentsline {subsection}{\numberline {1.3.2}Unbounded Rucksackproblem}{3}{subsection.1.3.2}% -\contentsline {subsection}{\numberline {1.3.3}Multiples Rucksackproblem}{3}{subsection.1.3.3}% -\contentsline {chapter}{\numberline {2}Beispiel Code-schnipsel einbinden}{4}{chapter.2}% -\contentsline {section}{\numberline {2.1}lorem ipsum}{4}{section.2.1}% -\contentsline {section}{\numberline {2.2}Verweis auf Code}{5}{section.2.2}% -\contentsline {chapter}{Glossar}{6}{section*.7}% -\contentsline {chapter}{\nonumberline Anhang}{7}{chapter*.9}% +\contentsline {subsection}{\numberline {1.3.3}Multiples Rucksackproblem}{4}{subsection.1.3.3}% +\contentsline {chapter}{\numberline {2}Beispiel Code-schnipsel einbinden}{5}{chapter.2}% +\contentsline {section}{\numberline {2.1}lorem ipsum}{5}{section.2.1}% +\contentsline {section}{\numberline {2.2}Verweis auf Code}{6}{section.2.2}% +\contentsline {chapter}{\nonumberline Literatur}{7}{chapter*.8}% +\contentsline {chapter}{Glossar}{8}{section*.9}% +\contentsline {chapter}{\nonumberline Anhang}{9}{chapter*.11}% \providecommand \tocbasic@end@toc@file {}\tocbasic@end@toc@file diff --git a/sources/sources.bib b/sources/sources.bib deleted file mode 100644 index 036b434..0000000 --- a/sources/sources.bib +++ /dev/null @@ -1,106 +0,0 @@ -% Books -@book{Luderer2017, - title={Facetten der Wirtschaftsmathematik: Eine unterhaltsame Einf{\"u}hrung ganz ohne Formeln}, - author={Luderer, B.}, - isbn={9783658191870}, - url={https://books.google.de/books?id=F256swEACAAJ}, - year={2017}, - pages = {104-105}, - publisher={Springer Fachmedien Wiesbaden} -} - -@article{Chu1998, - title={A Genetic Algorithm for the Multidimensional Knapsack Problem}, - author={Chu, P. and Beasley, J.}, - journal={Journal of Heuristics}, - volume={4}, - number={1}, - pages={63-86}, - year={1998}, - doi={10.1023/A:1009642405419}, - publisher={Springer} -} - -@book{Vocking2008, - title={Taschenbuch der Algorithmen}, - editor={Vöcking, B. and weitere}, - year={2008}, - publisher={Springer}, - address={Berlin, Heidelberg}, - pages={405-413}, - isbn={9783540763932} -} - -@incollection{Martello1987, - title = {Algorithms for Knapsack Problems}, - editor = {Silvano Martello and Gilbert Laporte and Michel Minoux and Celso Ribeiro}, - series = {North-Holland Mathematics Studies}, - publisher = {North-Holland}, - volume = {132}, - pages = {213-257}, - year = {1987}, - booktitle = {Surveys in Combinatorial Optimization}, - issn = {0304-0208}, - doi = {https://doi.org/10.1016/S0304-0208(08)73237-7}, - url = {https://www.sciencedirect.com/science/article/pii/S0304020808732377}, - author = {Silvano Martello and Paolo Toth}, - abstract = {Publisher Summary - This chapter presents a great variety of practical problems that are represented by a set of entities, each having an associated value, from which one or more subsets has to be selected in such a way that the sum of the values of the selected entities is maximized and some predefined conditions. The most common condition is obtained by associating a size to each entity and establishing that the sum of the entity sizes in each subset does not exceed some prefixed bound. These problems are generally called knapsack problems, as they recall the situation of a hitch-hiker having to fill up knapsack by selecting from among various possible objects those that provides the maximum comfort. The entities are called as items and their number is indicated by n. The value and size associated with the j-th item are called as profit and weight. The majority of problems are considered as single knapsack problems, where one container must be filled with an optimal subset of items. The chapter reviews both exact and approximate algorithms for the solution of the most important types of knapsack problems. Exact solutions are obtained through branch-and-bound algorithms or dynamic programming, or a combination of the two approaches. The performances of these algorithms are evaluated by computational experiments with randomly generated test problems. Approximate algorithms are of the greedy type or are based on scaling. They are evaluated on the basis of their worst-case performance.} -} - -@INPROCEEDINGS{Lawler1977, - author={Lawler, E. L.}, - booktitle={18th Annual Symposium on Foundations of Computer Science (sfcs 1977)}, - title={Fast approximation algorithms for knapsack problems}, - year={1977}, - volume={}, - number={}, - pages={206-213}, - doi={10.1109/SFCS.1977.11}} - - -@mastersthesis{Schury2013, - title={Social Choice Methoden zur Lösung von mehrdimensionalen Rucksackproblemen / Social Choice Methods for the solution of the multimensional knapsack problem}, - author={Schury, Christoph}, - year={2013}, - school={Universität Graz}, - language={German}, - keywords={Rucksackproblem, Kollektiventscheidung, Online-Publikation}, - urn={urn:nbn:at:at-ubg:1-47652}, - pages={IV, 55}, - advisor={Pferschy, Ulrich} -} - - -@Inbook{Domschke2015, -author="Domschke, Wolfgang -and Drexl, Andreas -and Klein, Robert -and Scholl, Armin", -title="Ganzzahlige und kombinatorische Optimierung", -bookTitle="Einf{\"u}hrung in Operations Research", -year="2015", -publisher="Springer Berlin Heidelberg", -address="Berlin, Heidelberg", -pages="127--164", -abstract="Die meisten der in den Kapiteln 2 bis 5 betrachteten Probleme sind als lineare Optimierungsprobleme formulierbar und mit dem Simplex-Algorithmus oder spezialisierten Vorgehensweisen l{\"o}sbar. Wesentliche Eigenschaft dieser Probleme ist neben der Linearit{\"a}t von Zielfunktion und Nebenbedingungen, dass ausschlie{\ss}lich kontinuierliche Variablen vorkommen. Im Gegensatz dazu wenden wir uns nun einer Klasse von Problemen zu, bei der auch bin{\"a}re oder ganzzahlige Variablen zugelassen sind. Speziell durch die Verwendung von Bin{\"a}rvariablen ergibt sich die M{\"o}glichkeit, auch komplexere Wirkungszusammenh{\"a}nge wie etwa logische Bedingungen zu modellieren. So beruht die Formulierung des Grundmodells der Kapazit{\"a}tsplanung im Projektmanagement (vgl. Kap. 5.5) auf dem Einsatz von Bin{\"a}rvariablen. Weitere Einsatzgebiete der ganzzahligen und kombinatorischen Optimierung umfassen die Ablaufplanung, die Logistik und die Personaleinsatzplanung (vgl. auch die Beispiele in Kap. 1.4).", -isbn="978-3-662-48216-2", -doi="10.1007/978-3-662-48216-2_6", -url="https://doi.org/10.1007/978-3-662-48216-2_6" -} - - -@article{Martello1990, -title = {An exact algorithm for large unbounded knapsack problems}, -journal = {Operations Research Letters}, -volume = {9}, -number = {1}, -pages = {15-20}, -year = {1990}, -issn = {0167-6377}, -doi = {https://doi.org/10.1016/0167-6377(90)90035-4}, -url = {https://www.sciencedirect.com/science/article/pii/0167637790900354}, -author = {Silvano Martello and Paolo Toth}, -keywords = {integer programming, knapsack problem, branch-and-bound}, -abstract = {Given n types, each having an associated profit and weight, and a container of given capacity, the unbounded knapsack problem is to determine the number of items of each type to be selected so that the corresponding total profit is a maximum and the corresponding total weight does not exceed the capacity. We present upper bounds, dominance relations, and an approach—based on the definition of a core problem—to the exact solution of very large instances of the problem. We give the results of computational experiments on randomly generated test problems involving up to 250 000 item types.} -}