From f6a0d0deff519d736aae64d07c12890693b2ccdb Mon Sep 17 00:00:00 2001
From: dschlueter <dieter.schlueter@linix.de>
Date: Wed, 9 Sep 2026 23:09:40 +0200
Subject: [PATCH] =?UTF-8?q?Vorwort:=20Hinweis=20zur=20Entstehung=20und=20v?=
 =?UTF-8?q?erwendeten=20Werkzeugen=20erg=C3=A4nzt?=
MIME-Version: 1.0
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---
 ...rations_Research_mit_Python_Version_04.pdf | Bin 5723628 -> 5725588 bytes
 OR_HTML_04/assets/search-index.js             |   2 +-
 OR_HTML_04/gesamtdokument.html                |   5 ++++-
 OR_HTML_04/vorwort.html                       |   4 +++-
 Operations_Research_mit_Python_Version_04.md  |   5 +++++
 Operations_Research_mit_Python_Version_04.pdf | Bin 5723628 -> 5725588 bytes
 .../00_Vorwort_und_Lesehilfe.md               |   6 ++++++
 7 files changed, 19 insertions(+), 3 deletions(-)

diff --git a/OR_HTML_04/Operations_Research_mit_Python_Version_04.pdf b/OR_HTML_04/Operations_Research_mit_Python_Version_04.pdf
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Die Symbole der Optimierung","anker":"die-symbole-der-optimierung","text":""},{"seite":"notation.html","titel":"1.1 Grundgrößen jedes Modells","anker":"grundgrößen-jedes-modells","text":""},{"seite":"notation.html","titel":"1.2 Mengen und Zahlbereiche","anker":"mengen-und-zahlbereiche","text":""},{"seite":"notation.html","titel":"1.3 Operatoren und Schreibweisen","anker":"operatoren-und-schreibweisen","text":""},{"seite":"notation.html","titel":"1.4 Symbole der Finanzkapitel (Teil IV)","anker":"symbole-der-finanzkapitel-teil-iv","text":""},{"seite":"notation.html","titel":"1.5 Symbole der Verfahrenskapitel (Teil II und III)","anker":"symbole-der-verfahrenskapitel-teil-ii-und-iii","text":"Diese Zeichen kommen erst in den Kapiteln vor, die über das klassische LP hinausgehen."},{"seite":"notation.html","titel":"2. 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Skalare bleiben mager: n, Z."},{"seite":"einfuehrung.html","titel":"Kapitel 1: Einführung in Operations Research — Vom Ursprung zur mathematischen Entscheidungsfindung","anker":null,"text":""},{"seite":"einfuehrung.html","titel":"1.1 In 5 Minuten gelöst","anker":"sec:einfuehrung-schnellstart","text":"Bevor wir über Theorie sprechen, lösen Sie ein echtes Problem. Kopieren Sie den folgenden Block, führen Sie ihn aus, sehen Sie das Ergebnis — die Erklärung kommt danach."},{"seite":"einfuehrung.html","titel":"1.2 Lernziele","anker":"sec:einfuehrung-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"einfuehrung.html","titel":"1.3 Was ist Operations Research wirklich?","anker":"sec:einfuehrung-was-ist-operations-research-wirklich","text":"Im Alltag lösen wir Probleme intuitiv: durch Ausprobieren, Daumenregeln, Erfahrung. Das funktioniert erstaunlich gut — bis eine bestimmte Schwelle überschritten wird. Dann bricht d"},{"seite":"einfuehrung.html","titel":"Die drei Stufen der Analytik","anker":"die-drei-stufen-der-analytik","text":"OR beantwortet nicht die beschreibende Frage „Was ist passiert?“ und nicht die vorhersagende Frage „Was wird passieren?“, sondern die vorschreibende (präskriptive) Kernfrage:"},{"seite":"einfuehrung.html","titel":"1.4 Warum Ausprobieren scheitert — mit eigener Rechnung","anker":"sec:einfuehrung-warum-ausprobieren-scheitert-mit-eigener-rechnung","text":"Die Behauptung „vollständiges Durchprobieren ist unmöglich“ wird oft aufgestellt und selten belegt. Rechnen wir sie einmal selbst nach — das Ergebnis prägt sich besser ein als jede"},{"seite":"einfuehrung.html","titel":"1.5 Historischer Kontext und Evolution","anker":"sec:einfuehrung-historischer-kontext-und-evolution","text":"Der Begriff entstand im Vorfeld des Zweiten Weltkriegs in Großbritannien. Militärische Führungsstäbe standen vor neuartigen logistischen und strategischen Fragen: Wie platziert man"},{"seite":"einfuehrung.html","titel":"1.6 Die vier universellen Bausteine jedes OR-Problems","anker":"sec:einfuehrung-die-vier-universellen-bausteine-jedes-or","text":"Dies ist der wichtigste Abschnitt des Kapitels. Jedes Optimierungsproblem der Welt lässt sich — unabhängig von der Branche — auf vier Elemente reduzieren. Wenn Sie später vor einem"},{"seite":"einfuehrung.html","titel":"Baustein 1 — Entscheidungsvariablen (x)","anker":"baustein-1-entscheidungsvariablen-x","text":"Die Stellschrauben des Modells: Werte, die der Algorithmus aktiv festlegen darf."},{"seite":"einfuehrung.html","titel":"Baustein 2 — Parameter / Eingabedaten (c, A, b)","anker":"baustein-2-parameter-eingabedaten-c-a-b","text":"Die unveränderlichen Fakten der Realität: historische Renditen, Maschinenstundensätze, Lagerkapazitäten, Steuerquoten, Entfernungen, Qualifikationen."},{"seite":"einfuehrung.html","titel":"Baustein 3 — Zielfunktion (f(x))","anker":"baustein-3-zielfunktion-fx","text":"Ein mathematischer Ausdruck, der die Güte einer Lösung zu einer einzigen Zahl verdichtet:"},{"seite":"einfuehrung.html","titel":"Baustein 4 — Nebenbedingungen (Constraints)","anker":"baustein-4-nebenbedingungen-constraints","text":"Regeln, die den Raum der erlaubten Lösungen einschränken:"},{"seite":"einfuehrung.html","titel":"Die vier Bausteine als eine Formel","anker":"die-vier-bausteine-als-eine-formel","text":"Setzt man die vier Bausteine zusammen, entsteht die Standardform, in der Ihnen jedes Optimierungsmodell dieses Buches begegnen wird. Sie sieht auf den ersten Blick sperrig aus — de"},{"seite":"einfuehrung.html","titel":"Die Bausteine in der Praxis: eine Vorlage","anker":"die-bausteine-in-der-praxis-eine-vorlage","text":"Damit die Zerlegung zur Gewohnheit wird, hier eine Vorlage, die Sie bei jedem neuen Problem ausfüllen können — vor der ersten Codezeile."},{"seite":"einfuehrung.html","titel":"1.7 Erstes Python-Vollbeispiel: Ressourcenallokation im Rechenzentrum","anker":"sec:einfuehrung-erstes-python-vollbeispiel-ressourcenallokation-im-rechenzentrum","text":""},{"seite":"einfuehrung.html","titel":"Szenario","anker":"szenario","text":"Ein quantitatives Handelshaus betreibt Serverkapazitäten für zwei Algorithmentypen:"},{"seite":"einfuehrung.html","titel":"Mathematische Formulierung","anker":"mathematische-formulierung","text":"Entscheidungsvariablen:"},{"seite":"einfuehrung.html","titel":"Umsetzung mit Google OR-Tools","anker":"umsetzung-mit-google-or-tools","text":""},{"seite":"einfuehrung.html","titel":"1.8 Von Excel zu Python: Ihre Daten liegen schon da","anker":"sec:einfuehrung-excel-bruecke","text":"In den allermeisten Betrieben liegen die Zahlen, die ein Optimierungsmodell braucht, nicht in einer Datenbank, sondern in einer Tabellenkalkulation. Und viele Leserinnen und Leser "},{"seite":"einfuehrung.html","titel":"1.9 Übungsaufgaben","anker":"sec:einfuehrung-uebungsaufgaben","text":""},{"seite":"einfuehrung.html","titel":"1.10 Finde den Denkfehler","anker":"sec:einfuehrung-denkfehler","text":"Ein Programm, das abstürzt, ist harmlos — Sie sehen sofort, dass etwas nicht stimmt. Gefährlich sind Programme, die fehlerfrei durchlaufen, einen Status OPTIMAL melden und eine Zah"},{"seite":"einfuehrung.html","titel":"1.11 Micro-Quiz","anker":"sec:einfuehrung-quiz","text":""},{"seite":"einfuehrung.html","titel":"1.12 Selbsttest","anker":"sec:einfuehrung-selbsttest","text":""},{"seite":"einfuehrung.html","titel":"1.13 Zusammenfassung","anker":"sec:einfuehrung-zusammenfassung","text":""},{"seite":"fundament.html","titel":"Kapitel 2: Das mathematische Fundament — Vektoren, Matrizen, Konvexität","anker":null,"text":""},{"seite":"fundament.html","titel":"2.1 In 5 Minuten gelöst","anker":"sec:fundament-schnellstart","text":"Dieses Kapitel handelt von Geometrie. Bevor wir Vektoren und Matrizen einführen, sehen Sie die zentrale Einsicht in Aktion — ganz ohne Solver, mit acht Zeilen NumPy."},{"seite":"fundament.html","titel":"2.2 Lernziele","anker":"sec:fundament-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"fundament.html","titel":"2.3 Warum überhaupt Vektoren und Matrizen?","anker":"sec:fundament-warum-ueberhaupt-vektoren-und-matrizen","text":"In Kapitel 1 hatten wir zwei Variablen und drei Nebenbedingungen. Das ließ sich bequem ausschreiben. Reale Modelle haben hunderte bis hunderttausende Variablen — dort wäre Ausschre"},{"seite":"fundament.html","titel":"Die Bausteine","anker":"die-bausteine","text":"Sei n die Anzahl der Entscheidungsvariablen und m die Anzahl der Nebenbedingungen."},{"seite":"fundament.html","titel":"Die kanonische Standardform","anker":"die-kanonische-standardform","text":"Damit lässt sich jedes lineare Programm in drei Zeilen schreiben:"},{"seite":"fundament.html","titel":"Dasselbe in Python","anker":"dasselbe-in-python","text":""},{"seite":"fundament.html","titel":"2.4 Der zulässige Lösungsraum und das Polyeder","anker":"sec:fundament-der-zulaessige-loesungsraum-und-das-polyeder","text":"Jede lineare Ungleichung a_{i1}x_1 + a_{i2}x_2 \\le b_i definiert geometrisch eine Halbebene (im \\mathbb{R}^2) bzw. einen Halbraum (im \\mathbb{R}^n): Die zugehörige Gleichung ist ei"},{"seite":"fundament.html","titel":"Der Fundamentalsatz der linearen Optimierung","anker":"der-fundamentalsatz-der-linearen-optimierung","text":""},{"seite":"fundament.html","titel":"2.5 Konvexität: die Grenze zwischen leicht und schwer","anker":"sec:fundament-konvexitaet-die-grenze-zwischen-leicht-und","text":"Hier kommt die vielleicht wichtigste Einsicht des ganzen Buches. In der Optimierungstheorie verläuft die Trennlinie zwischen „zuverlässig lösbar“ und „im Allgemeinen hoffnungslos“ "},{"seite":"fundament.html","titel":"Konvexe Menge","anker":"konvexe-menge","text":"Eine Menge \\mathcal{C} \\subseteq \\mathbb{R}^n heißt konvex, wenn für alle Punkte \\mathbf{x}, \\mathbf{y} \\in \\mathcal{C} und jedes \\theta \\in [0,1] gilt:"},{"seite":"fundament.html","titel":"Konvexe Funktion","anker":"konvexe-funktion","text":"Eine Funktion f: \\mathcal{C} \\to \\mathbb{R} heißt konvex, wenn für alle \\mathbf{x}, \\mathbf{y} \\in \\mathcal{C} und \\theta \\in [0,1] gilt:"},{"seite":"fundament.html","titel":"Der zentrale Satz","anker":"der-zentrale-satz","text":""},{"seite":"fundament.html","titel":"Was ist konvex, was nicht?","anker":"was-ist-konvex-was-nicht","text":""},{"seite":"fundament.html","titel":"Konvexität sichtbar machen","anker":"konvexität-sichtbar-machen","text":""},{"seite":"fundament.html","titel":"2.6 Geometrische Visualisierung des Lösungsraums","anker":"sec:fundament-geometrische-visualisierung-des-loesungsraums","text":"Wir zeichnen nun den Lösungsraum aus Kapitel 1 und lesen das Optimum ab. Zweidimensionale Bilder sind der schnellste Weg, Intuition für höhere Dimensionen aufzubauen — auch wenn ma"},{"seite":"fundament.html","titel":"2.7 Wenn die Zahlen nicht zusammenpassen: Kondition und Skalierung","anker":"sec:fundament-kondition","text":"Bis hierher war die lineare Algebra exakt. Ein Rechner rechnet aber nicht exakt, sondern mit rund 16 signifikanten Stellen — und Ihre Betriebsdaten sind ohnehin nur auf wenige Stel"},{"seite":"fundament.html","titel":"Die geometrische Vorstellung: zwei fast parallele Linien","anker":"die-geometrische-vorstellung-zwei-fast-parallele-linien","text":"Zwei Nebenbedingungen legen im Zweidimensionalen einen Schnittpunkt fest. Stehen die zugehörigen Geraden kreuzweise aufeinander, ist der Schnittpunkt robust: Verschiebt man eine Ge"},{"seite":"fundament.html","titel":"Woher schlechte Kondition im Alltag kommt","anker":"woher-schlechte-kondition-im-alltag-kommt","text":"Fast nie aus exotischer Mathematik — fast immer aus Einheiten:"},{"seite":"fundament.html","titel":"Ruiz-Equilibrierung: das Modell gesundrechnen","anker":"ruiz-equilibrierung-das-modell-gesundrechnen","text":"Die Gegenmaßnahme ist erstaunlich schlicht. Man multipliziert jede Zeile und jede Spalte der Matrix mit einem Faktor, sodass die Beträge überall in derselben Größenordnung landen:"},{"seite":"fundament.html","titel":"Das Experiment","anker":"das-experiment","text":"Das folgende Programm führt beides vor — die Fehlerverstärkung und ihre Behebung — und schließt mit der praktischen Konsequenz, die Sie in Kapitel 6 wieder brauchen werden: warum e"},{"seite":"fundament.html","titel":"2.8 Übungsaufgaben","anker":"sec:fundament-uebungsaufgaben","text":""},{"seite":"fundament.html","titel":"2.9 Finde den Denkfehler","anker":"sec:fundament-denkfehler","text":""},{"seite":"fundament.html","titel":"2.10 Micro-Quiz","anker":"sec:fundament-quiz","text":""},{"seite":"fundament.html","titel":"2.11 Selbsttest","anker":"sec:fundament-selbsttest","text":""},{"seite":"fundament.html","titel":"2.12 Zusammenfassung","anker":"sec:fundament-zusammenfassung","text":""},{"seite":"oekosystem.html","titel":"Kapitel 3: Das Python-Ökosystem für OR — Solver, Bindings und Modellierungsschichten","anker":null,"text":""},{"seite":"oekosystem.html","titel":"3.1 In 5 Minuten gelöst","anker":"sec:oekosystem-schnellstart","text":""},{"seite":"oekosystem.html","titel":"3.2 Lernziele","anker":"sec:oekosystem-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"oekosystem.html","titel":"3.3 Die Zwei-Schichten-Architektur","anker":"sec:oekosystem-die-zwei-schichten-architektur","text":"Im modernen Operations Research schreibt man Optimierungsalgorithmen nicht selbst. Man nutzt eine zweischichtige Architektur:"},{"seite":"oekosystem.html","titel":"3.4 Die Werkzeuge im Vergleich","anker":"sec:oekosystem-die-werkzeuge-im-vergleich","text":""},{"seite":"oekosystem.html","titel":"Die Entscheidung in drei Fragen","anker":"die-entscheidung-in-drei-fragen","text":"Statt die Tabelle auswendig zu lernen, beantworten Sie drei Fragen:"},{"seite":"oekosystem.html","titel":"3.5 Ein System — vier Programmieransätze","anker":"sec:oekosystem-ein-system-vier-programmieransaetze","text":"Nichts macht die Unterschiede so deutlich wie dasselbe Problem, viermal gelöst. Wir nehmen:"},{"seite":"oekosystem.html","titel":"Wie die Isolation aussieht, wenn sie tragen soll","anker":"wie-die-isolation-aussieht-wenn-sie-tragen-soll","text":"„Eigener Prozess” ist schnell gesagt. Die naheliegende Umsetzung — ein Codeschnipsel als Zeichenkette an python -c übergeben — funktioniert und ist trotzdem die schlechteste: Der S"},{"seite":"oekosystem.html","titel":"Was das CSR-Format bedeutet","anker":"was-das-csr-format-bedeutet","text":"highspy erwartet die Nebenbedingungsmatrix im CSR-Format (Compressed Sparse Row, komprimierte Zeilendarstellung). Statt der vollen Matrix speichert man nur die Einträge ungleich nu"},{"seite":"oekosystem.html","titel":"3.6 Wann lohnt sich welche Ebene?","anker":"sec:oekosystem-wann-lohnt-sich-welche-ebene","text":""},{"seite":"oekosystem.html","titel":"3.7 Modellierungsschichten für große Modelle: Pyomo und Linopy","anker":"sec:oekosystem-pyomo-linopy","text":"Die vier Bibliotheken aus dem Vierfach-Vergleich decken den Alltag weitgehend ab. Sobald Modelle industrielle Größe erreichen — zehntausende Variablen, Daten aus mehreren Systemen,"},{"seite":"oekosystem.html","titel":"Pyomo: die algebraische Denkweise","anker":"pyomo-die-algebraische-denkweise","text":"Pyomo ist im deutschsprachigen Raum der De-facto-Standard für große LP- und MILP-Modelle in Energiewirtschaft, Chemie und Logistik. Sein Kennzeichen: Es denkt in Mengen und Indizes"},{"seite":"oekosystem.html","titel":"Linopy: eine Zeile, zehntausend Nebenbedingungen","anker":"linopy-eine-zeile-zehntausend-nebenbedingungen","text":"Linopy verfolgt einen anderen Ansatz: Variablen sind beschriftete Arrays (xarray), keine indizierten Einzelobjekte. Damit wird aus"},{"seite":"oekosystem.html","titel":"Die Schichten im Überblick","anker":"die-schichten-im-überblick","text":""},{"seite":"oekosystem.html","titel":"3.8 Wo die Zeit wirklich hingeht: vektorisierte Modellgenerierung","anker":"sec:oekosystem-vektorisierung","text":"Eine der hartnäckigsten Fehlannahmen in Optimierungsprojekten lautet: „Wenn es zu langsam ist, brauchen wir einen besseren Solver.“ Messen Sie erst — oft stimmt das nicht."},{"seite":"oekosystem.html","titel":"Vier Stufen an einem Transportproblem","anker":"vier-stufen-an-einem-transportproblem","text":"Wir bauen dasselbe Transportproblem (m Werke, n Kunden, m \\cdot n Variablen) auf vier Arten auf:"},{"seite":"oekosystem.html","titel":"3.9 Übungsaufgaben","anker":"sec:oekosystem-uebungsaufgaben","text":""},{"seite":"oekosystem.html","titel":"3.10 Finde den Denkfehler","anker":"sec:oekosystem-denkfehler","text":""},{"seite":"oekosystem.html","titel":"3.11 Micro-Quiz","anker":"sec:oekosystem-quiz","text":""},{"seite":"oekosystem.html","titel":"3.12 Selbsttest","anker":"sec:oekosystem-selbsttest","text":""},{"seite":"oekosystem.html","titel":"3.13 Zusammenfassung","anker":"sec:oekosystem-zusammenfassung","text":""},{"seite":"modellierung.html","titel":"Kapitel 4: Vom Management-Wunsch zum Modell","anker":null,"text":""},{"seite":"modellierung.html","titel":"4.1 In 5 Minuten gelöst","anker":"sec:modellierung-schnellstart","text":""},{"seite":"modellierung.html","titel":"4.2 Lernziele","anker":"sec:modellierung-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"modellierung.html","titel":"4.3 Ein Satz ist noch kein Modell","anker":"sec:modellierung-vier-fragen","text":"Was aus einer Besprechung mitkommt, klingt meist so:"},{"seite":"modellierung.html","titel":"4.4 Derselbe Datensatz, fünf Modelle","anker":"sec:modellierung-fuenf-modelle","text":"Das Programm dieses Kapitels arbeitet die vier Fragen an einer Auftragsannahme ab: 14 Anfragen verlangen 800 Maschinenstunden, verfügbar sind 300."},{"seite":"modellierung.html","titel":"Teil 2: Ein Satz, drei Lesarten","anker":"teil-2-ein-satz-drei-lesarten","text":"„Stammkunden dürfen wir nicht verlieren.” Jeder im Raum nickt. Niemand merkt, dass drei verschiedene Modelle gemeint sein können:"},{"seite":"modellierung.html","titel":"Teil 3: Hart oder weich","anker":"teil-3-hart-oder-weich","text":"Dieselbe Regel als Strafkosten je abgelehntem Stammkundenauftrag:"},{"seite":"modellierung.html","titel":"Teil 4: Ein Quartal später","anker":"teil-4-ein-quartal-später","text":"Die harte Lesart A wurde eingebaut, alles lief drei Monate lang. Dann fragt Weber einen Auftrag über 320 Maschinenstunden an — mehr, als der ganze Zeitraum hergibt."},{"seite":"modellierung.html","titel":"4.5 Der Gesprächsleitfaden","anker":"sec:modellierung-leitfaden","text":"Was aus den vier Teilen als Handwerkszeug bleibt — Fragen, die man in der ersten Besprechung stellt, bevor eine Zeile Code entsteht:"},{"seite":"modellierung.html","titel":"4.6 Deutsch — OR: ein Übersetzungslexikon","anker":"sec:modellierung-glossar","text":"Die zwölf Fragen liefern Sätze. Sätze sind noch kein Modell — aber die meisten von ihnen haben eine Standardübersetzung, und wer sie kennt, spart sich das Nachdenken für die Fälle,"},{"seite":"modellierung.html","titel":"Fünf Wendungen, bei denen die Übersetzung eine Entscheidung erzwingt","anker":"fünf-wendungen-bei-denen-die-übersetzung-eine-entscheidung-erzwingt","text":"Die Tabelle oben tut so, als sei die Zuordnung eindeutig. Bei den folgenden fünf ist sie es nicht — und das Nachfragen ist die eigentliche Modellierungsarbeit:"},{"seite":"modellierung.html","titel":"Und die Gegenrichtung: OR — Deutsch","anker":"und-die-gegenrichtung-or-deutsch","text":"Dieselbe Barriere gibt es rückwärts. Diese Sätze haben sich bewährt, wenn ein Ergebnis jemandem erklärt werden muss, der das Modell nicht kennt:"},{"seite":"modellierung.html","titel":"4.7 Übungsaufgaben","anker":"sec:modellierung-uebungsaufgaben","text":""},{"seite":"modellierung.html","titel":"4.8 Finde den Denkfehler","anker":"sec:modellierung-denkfehler","text":""},{"seite":"modellierung.html","titel":"4.9 Micro-Quiz","anker":"sec:modellierung-quiz","text":""},{"seite":"modellierung.html","titel":"4.10 Selbsttest","anker":"sec:modellierung-selbsttest","text":""},{"seite":"modellierung.html","titel":"4.11 Zusammenfassung","anker":"sec:modellierung-zusammenfassung","text":""},{"seite":"synthese-grundlagen.html","titel":"Synthese Teil I","anker":null,"text":""},{"seite":"synthese-grundlagen.html","titel":"Was welches Kapitel klärt","anker":"was-welches-kapitel-klärt","text":""},{"seite":"synthese-grundlagen.html","titel":"Die Reihenfolge ist die Aussage","anker":"die-reihenfolge-ist-die-aussage","text":"Dieser Teil steht bewusst vor den Verfahren. Die drei häufigsten Gründe, aus denen ein OR-Projekt scheitert, liegen alle hier — und keiner davon ist ein Solverproblem:"},{"seite":"synthese-grundlagen.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"lp.html","titel":"Kapitel 5: Lineare Programmierung — Simplex, Dualität und Schattenpreise","anker":null,"text":""},{"seite":"lp.html","titel":"5.1 In 5 Minuten gelöst","anker":"sec:lp-schnellstart","text":"Bisher hat ein Solver Ihnen gesagt, was Sie tun sollen. Jetzt sagt er Ihnen zusätzlich, was eine zusätzliche Stunde wert wäre — und das ist die Zahl, für die Sie im Betrieb gehört "},{"seite":"lp.html","titel":"5.2 Lernziele","anker":"sec:lp-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"lp.html","titel":"5.3 Die Standardform und Schlupfvariablen","anker":"sec:lp-die-standardform-und-schlupfvariablen","text":"Lineare Programme zeichnen sich dadurch aus, dass Zielfunktion und alle Nebenbedingungen strikt linear sind — keine Produkte von Variablen, keine Quadrate, keine Logarithmen."},{"seite":"lp.html","titel":"5.4 Der Simplex-Algorithmus Schritt für Schritt","anker":"sec:lp-der-simplex-algorithmus-schritt-fuer-schritt","text":"Der von George Dantzig 1947 formulierte Simplex-Algorithmus nutzt den Fundamentalsatz aus Kapitel 2: Das Optimum liegt in einer Ecke. Statt alle Ecken aufzuzählen, wandert er von E"},{"seite":"lp.html","titel":"Der Ablauf","anker":"der-ablauf","text":""},{"seite":"lp.html","titel":"5.5 Das Simplex-Tableau in Python","anker":"sec:lp-das-simplex-tableau-in-python","text":""},{"seite":"lp.html","titel":"5.6 Dualität und Schattenpreise","anker":"sec:lp-dualitaet-und-schattenpreise","text":"Zu jedem linearen Optimierungsproblem (dem primalen Problem) existiert ein spiegelbildliches Zwillingsproblem: das duale Problem."},{"seite":"lp.html","titel":"Der starke Dualitätssatz","anker":"der-starke-dualitätssatz","text":""},{"seite":"lp.html","titel":"Der Schattenpreis als Ableitung","anker":"der-schattenpreis-als-ableitung","text":"Der Dualwert y_i^* ist die Grenzproduktivität der Ressource i:"},{"seite":"lp.html","titel":"5.7 Die Vorzeichenfalle bei Schattenpreisen","anker":"sec:lp-die-vorzeichenfalle-bei-schattenpreisen","text":"Diese Falle sollten Sie sich merken, weil sie in jedem Projekt wieder auftaucht."},{"seite":"lp.html","titel":"5.8 Praxisfall: Sensitivitätsanalyse mit korrekten Schattenpreisen","anker":"sec:lp-praxisfall-sensitivitaetsanalyse-mit-korrekten-schattenpreisen","text":"Szenario. Ein Fertigungsbetrieb stellt drei Produkte her und will wissen:"},{"seite":"lp.html","titel":"5.9 Wann Schattenpreise lügen: Entartung und Toleranzen","anker":"sec:lp-entartung","text":"Der vorige Abschnitt hat gezeigt, wie wertvoll Schattenpreise sind. Dieser zeigt die zwei Fälle, in denen sie eine Entscheidung nicht tragen — und wie man beide erkennt, bevor jema"},{"seite":"lp.html","titel":"Fall 1: Entartung","anker":"fall-1-entartung","text":"Erinnern Sie sich an Abschnitt 2.4: In zwei Dimensionen legen zwei sich schneidende Geraden eine Ecke fest, in drei Dimensionen drei Ebenen, allgemein n Restriktionen bei n Variabl"},{"seite":"lp.html","titel":"Fall 2: Toleranzen","anker":"fall-2-toleranzen","text":"Der zweite Fall ist unscheinbarer und deshalb häufiger. Wie stellt man fest, ob eine Ressource ausgelastet ist? Naheliegend wäre if schlupf == 0. Genau das ist falsch."},{"seite":"lp.html","titel":"Die ehrliche Auskunft: Schattenpreis-Spannen","anker":"die-ehrliche-auskunft-schattenpreis-spannen","text":"Was tut man also, wenn das Modell entartet ist? Man meldet keinen Einzelwert, sondern die Spanne. Die Menge aller optimalen Dualvektoren ist selbst ein Polyeder:"},{"seite":"lp.html","titel":"5.10 Übungsaufgaben","anker":"sec:lp-uebungsaufgaben","text":""},{"seite":"lp.html","titel":"5.11 Finde den Denkfehler","anker":"sec:lp-denkfehler","text":""},{"seite":"lp.html","titel":"5.12 Micro-Quiz","anker":"sec:lp-quiz","text":""},{"seite":"lp.html","titel":"5.13 Selbsttest","anker":"sec:lp-selbsttest","text":""},{"seite":"lp.html","titel":"5.14 Zusammenfassung","anker":"sec:lp-zusammenfassung","text":""},{"seite":"milp.html","titel":"Kapitel 6: Gemischt-ganzzahlige Optimierung — Diskrete Entscheidungen und Branch-and-Bound","anker":null,"text":""},{"seite":"milp.html","titel":"6.1 In 5 Minuten gelöst","anker":"sec:milp-schnellstart","text":""},{"seite":"milp.html","titel":"6.2 Lernziele","anker":"sec:milp-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"milp.html","titel":"6.3 Warum Runden fundamental scheitert","anker":"sec:milp-warum-runden-fundamental-scheitert","text":"In der realen Welt sind viele Entscheidungen nicht teilbar. Man kann nicht 0,47 Flugzeuge kaufen, keine halbe Lagerhalle bauen, keinen Mitarbeiter zu 38 % einstellen. Noch wichtige"},{"seite":"milp.html","titel":"6.4 Branch-and-Bound","anker":"sec:milp-branch-and-bound","text":"MILP-Probleme sind NP-schwer. Der Standardansatz moderner Solver (HiGHS, SCIP, Gurobi) heißt Branch-and-Cut — Branch-and-Bound plus Schnittebenen."},{"seite":"milp.html","titel":"Die drei Phasen","anker":"die-drei-phasen","text":""},{"seite":"milp.html","titel":"6.5 Modellierungstricks: Big-M und logische Bedingungen","anker":"sec:milp-modellierungstricks-big-m-und-logische-bedingungen","text":"Hier kommt das Handwerkszeug, das MILP so mächtig macht: Man übersetzt Geschäftsregeln in lineare Ungleichungen — mithilfe binärer Hilfsvariablen y \\in \\{0,1\\} und einer hinreichen"},{"seite":"milp.html","titel":"Muster 1 — Fixkosten / Aktivierungsschalter","anker":"muster-1-fixkosten-aktivierungsschalter","text":"Regel: Wird Maschine j genutzt (x_j &gt; 0), fallen Rüstkosten F_j an."},{"seite":"milp.html","titel":"Muster 2 — Entweder-Oder (disjunctive constraints)","anker":"muster-2-entweder-oder-disjunctive-constraints","text":"Regel: Es muss entweder f(\\mathbf{x}) \\le b_1 oder g(\\mathbf{x}) \\le b_2 gelten."},{"seite":"milp.html","titel":"Muster 3 — Wenn-Dann (Implikation)","anker":"muster-3-wenn-dann-implikation","text":"Regel: Wenn y_1 = 1, dann muss auch y_2 = 1 sein."},{"seite":"milp.html","titel":"Muster 4 — Kardinalität („höchstens K aus N“)","anker":"muster-4-kardinalität-höchstens-k-aus-n","text":"\\sum_{j=1}^N y_j \\le K"},{"seite":"milp.html","titel":"Muster 5 — Semikontinuierlich („entweder 0 oder mindestens L“)","anker":"muster-5-semikontinuierlich-entweder-0-oder-mindestens-l","text":"Regel: Eine Position ist entweder gar nicht besetzt oder mit mindestens L Euro."},{"seite":"milp.html","titel":"6.6 Beispiel: Das Rucksackproblem","anker":"sec:milp-beispiel-das-rucksackproblem","text":"Bevor wir zum Portfolio kommen, das klassische Einstiegsproblem — kurz, verständlich und überall wiederzuerkennen."},{"seite":"milp.html","titel":"6.7 Praxisfall: Portfolio mit Ordergebühren und Kardinalitätsgrenze","anker":"sec:milp-praxisfall-portfolio-mit-ordergebuehren-und-kardinalitaetsgrenze","text":"Szenario. Ein Investor verteilt 100\\,000\\,\\text{€} auf sechs Anlageklassen."},{"seite":"milp.html","titel":"6.8 Wenn der Solver nicht fertig wird: Gap, Zeitlimit und Warm-Start","anker":"sec:milp-gap","text":"Bei einem LP gibt es zwei Ausgänge: eine optimale Lösung oder eine klare Absage. Bei einem MILP gibt es einen dritten, und im Betrieb ist er der häufigste:"},{"seite":"milp.html","titel":"Die zwei Zahlen, die zählen","anker":"die-zwei-zahlen-die-zählen","text":"Branch-and-Bound führt zu jedem Zeitpunkt zwei Werte mit:"},{"seite":"milp.html","titel":"Alle Statusfälle behandeln","anker":"alle-statusfälle-behandeln","text":"Der zweite Teil der Professionalität ist unspektakulär, aber entscheidend: jeden Rückgabewert auswerten, nicht nur OPTIMAL."},{"seite":"milp.html","titel":"Warm-Starts: was sie können und was nicht","anker":"warm-starts-was-sie-können-und-was-nicht","text":"Ein Warm-Start gibt dem Solver eine bekannte Lösung als Startpunkt mit. Die Idee ist verlockend: Wer schon eine brauchbare Lösung hat, muss nicht bei null anfangen."},{"seite":"milp.html","titel":"6.9 Übungsaufgaben","anker":"sec:milp-uebungsaufgaben","text":""},{"seite":"milp.html","titel":"6.10 Finde den Denkfehler","anker":"sec:milp-denkfehler","text":"Die Warnung „M so klein wie möglich“ wird meist mit der Laufzeit begründet: schwache Relaxation, mehr Knoten. Das stimmt — und ist die harmlosere Hälfte der Wahrheit."},{"seite":"milp.html","titel":"6.11 Micro-Quiz","anker":"sec:milp-quiz","text":""},{"seite":"milp.html","titel":"6.12 Selbsttest","anker":"sec:milp-selbsttest","text":""},{"seite":"milp.html","titel":"6.13 Zusammenfassung","anker":"sec:milp-zusammenfassung","text":""},{"seite":"cpsat.html","titel":"Kapitel 7: Constraint Programming mit CP-SAT — Logik, Scheduling und Zuweisung","anker":null,"text":""},{"seite":"cpsat.html","titel":"7.1 In 5 Minuten gelöst","anker":"sec:cpsat-schnellstart","text":""},{"seite":"cpsat.html","titel":"7.2 Lernziele","anker":"sec:cpsat-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"cpsat.html","titel":"7.3 Ein anderes Denkmodell","anker":"sec:cpsat-ein-anderes-denkmodell","text":"Während LP und MILP auf algebraischen Gleichungen und Simplex-Matrizen beruhen, kommt Constraint Programming (CP) aus der künstlichen Intelligenz:"},{"seite":"cpsat.html","titel":"Was Propagation leistet","anker":"was-propagation-leistet","text":""},{"seite":"cpsat.html","titel":"7.4 Globale Constraints — die Bausteine von CP-SAT","anker":"sec:cpsat-globale-constraints-die-bausteine-von-cp","text":"CP-SAT stellt mächtige Makros bereit, die in MILP hunderte Big-M-Ungleichungen erfordern würden. Jedes davon hat einen spezialisierten, hocheffizienten Propagator."},{"seite":"cpsat.html","titel":"7.5 Praxisfall: Dynamisches Vertretungssystem","anker":"sec:cpsat-praxisfall-dynamisches-vertretungssystem","text":"Problemstellung. Eine Schule muss nach spontanen Ausfällen vier Unterrichtsstunden nachbesetzen."},{"seite":"cpsat.html","titel":"7.6 Intervallvariablen: Job-Shop-Scheduling","anker":"sec:cpsat-intervallvariablen-job-shop-scheduling","text":"Die Königsdisziplin von CP-SAT sind Zeitplanungsprobleme. Intervallvariablen sind dabei das zentrale Werkzeug."},{"seite":"cpsat.html","titel":"7.7 Parallele Suche: was num_workers wirklich bewirkt","anker":"sec:cpsat-parallele-suche","text":"An mehreren Stellen dieses Buchs steht loeser.parameters.num_workers = 1, mit dem Kommentar „für eine reproduzierbare Ausgabe“. Darin stecken zwei Behauptungen: dass parallele Such"},{"seite":"cpsat.html","titel":"Was die Messung zeigt","anker":"was-die-messung-zeigt","text":"Erstens: Die Beschleunigung ist überlinear. Acht Arbeiter sind im abgedruckten Lauf nicht achtmal, sondern gut zwölfmal schneller als einer. Das sieht nach Messfehler aus, ist aber"},{"seite":"cpsat.html","titel":"Was daraus für Tests folgt","anker":"was-daraus-für-tests-folgt","text":"Der Zielwert war in allen sechzehn Läufen derselbe: 183. Die Gestalt der Lösung war es nicht. Genau daran entscheidet sich, was ein Test prüfen darf:"},{"seite":"cpsat.html","titel":"7.8 Die fünf Antworten von CP-SAT","anker":"sec:cpsat-statusfaelle","text":"Abschnitt 6.8 hat gezeigt, warum OPTIMAL bei ganzzahligen Problemen nicht der Normalfall ist. Bei CP-SAT kommt eine Besonderheit hinzu: Es gibt fünf verschiedene Antworten, und dre"},{"seite":"cpsat.html","titel":"7.9 Übungsaufgaben","anker":"sec:cpsat-uebungsaufgaben","text":""},{"seite":"cpsat.html","titel":"7.10 Finde den Denkfehler","anker":"sec:cpsat-denkfehler","text":""},{"seite":"cpsat.html","titel":"7.11 Micro-Quiz","anker":"sec:cpsat-quiz","text":""},{"seite":"cpsat.html","titel":"7.12 Selbsttest","anker":"sec:cpsat-selbsttest","text":""},{"seite":"cpsat.html","titel":"7.13 Zusammenfassung","anker":"sec:cpsat-zusammenfassung","text":""},{"seite":"graphen.html","titel":"Kapitel 8: Graphen, Flüsse und Touren — Min-Cost-Flow, Matching und VRP","anker":null,"text":""},{"seite":"graphen.html","titel":"8.1 In 5 Minuten gelöst","anker":"sec:graphen-schnellstart","text":""},{"seite":"graphen.html","titel":"8.2 Lernziele","anker":"sec:graphen-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"graphen.html","titel":"8.3 Graphen als Modellsprache","anker":"sec:graphen-graphen-als-modellsprache","text":"Viele reale Probleme in Logistik, Kommunikation und Finanzströmen sind keine flachen Ungleichungssysteme, sondern Graphen G = (V, E):"},{"seite":"graphen.html","titel":"Das Minimum-Cost-Flow-Problem (MCNFP)","anker":"das-minimum-cost-flow-problem-mcnfp","text":"Minimum-Cost Network Flow Problem — deutsch: kostenminimales Flussproblem. Es ist das mathematische Fundament für Transportketten, Liquiditätsrouting und Datenverteilung."},{"seite":"graphen.html","titel":"8.4 Bipartites Matching: das Zuordnungsproblem","anker":"sec:graphen-bipartites-matching-das-zuordnungsproblem","text":"Wenn N Aufgaben auf N Ressourcen eins zu eins verteilt werden — Orders auf Broker, Schichten auf Mitarbeitende, Aufträge auf Maschinen — spricht man von bipartitem Matching."},{"seite":"graphen.html","titel":"Der Satz von Birkhoff und von Neumann","anker":"der-satz-von-birkhoff-und-von-neumann","text":""},{"seite":"graphen.html","titel":"8.5 Das Vehicle Routing Problem mit Zeitfenstern","anker":"sec:graphen-das-vehicle-routing-problem-mit-zeitfenstern","text":"Das Traveling Salesperson Problem (TSP), deutsch Problem des Handlungsreisenden, fragt nach der kürzesten Rundreise durch N Städte. Das Capacitated Vehicle Routing Problem with Tim"},{"seite":"graphen.html","titel":"Kurzzyklen verhindern","anker":"kurzzyklen-verhindern","text":"Ein naives Modell erlaubt Subtouren: isolierte Kreise, die das Depot nie anfahren. Die klassische Gegenmaßnahme ist die MTZ-Formulierung nach Miller, Tucker und Zemlin. Man führt R"},{"seite":"graphen.html","titel":"Praxisbeispiel: Flotten-Routing","anker":"praxisbeispiel-flotten-routing","text":""},{"seite":"graphen.html","titel":"8.6 Übungsaufgaben","anker":"sec:graphen-uebungsaufgaben","text":""},{"seite":"graphen.html","titel":"8.7 Finde den Denkfehler","anker":"sec:graphen-denkfehler","text":""},{"seite":"graphen.html","titel":"8.8 Micro-Quiz","anker":"sec:graphen-quiz","text":""},{"seite":"graphen.html","titel":"8.9 Selbsttest","anker":"sec:graphen-selbsttest","text":""},{"seite":"graphen.html","titel":"8.10 Zusammenfassung","anker":"sec:graphen-zusammenfassung","text":""},{"seite":"metaheuristiken.html","titel":"Kapitel 9: Metaheuristiken — wenn der exakte Solver aussteigt","anker":null,"text":""},{"seite":"metaheuristiken.html","titel":"9.1 In 5 Minuten gelöst","anker":"sec:metaheuristiken-schnellstart","text":""},{"seite":"metaheuristiken.html","titel":"9.2 Lernziele","anker":"sec:metaheuristiken-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"metaheuristiken.html","titel":"9.3 Die Aufgabe: Rüstzeiten an der Lackieranlage","anker":"sec:metaheuristiken-aufgabe","text":"Der Schnellstart war ein Spielzeug. Die Aufgabe dieses Kapitels ist dieselbe, nur in Betriebsgröße: n Aufträge in eine Reihenfolge bringen, so dass die Summe der Rüstzeiten minimal"},{"seite":"metaheuristiken.html","titel":"Die Instanz","anker":"die-instanz","text":"Alle Programme dieses Kapitels benutzen dieselbe Aufgabe. Die Rüstzeit setzt sich aus zwei Teilen zusammen: einem unregelmäßigen Reinigungsaufwand zwischen den Produktfamilien und "},{"seite":"metaheuristiken.html","titel":"9.4 Lokale Suche: der Zug und seine Kosten","anker":"sec:metaheuristiken-lokale-suche","text":"Eine lokale Suche braucht drei Dinge: eine Startlösung, einen Zug und eine Regel, welche Züge übernommen werden."},{"seite":"metaheuristiken.html","titel":"9.5 Simulated Annealing","anker":"sec:metaheuristiken-annealing","text":"Reines Bergsteigen — nur Verbesserungen annehmen — bleibt im ersten lokalen Optimum stehen. Simulated Annealing nimmt Verschlechterungen mit einer Wahrscheinlichkeit an, die von de"},{"seite":"metaheuristiken.html","titel":"Was die Messung zeigt — und was sie nicht zeigt","anker":"was-die-messung-zeigt-und-was-sie-nicht-zeigt","text":"Erstens: Der Beitrag des Annealings ist bescheiden. Bergsteigen kommt auf 917 Minuten, Annealing auf 907 — das simple Verfahren holt 93 % des Ertrags. Das ist kein Argument gegen A"},{"seite":"metaheuristiken.html","titel":"9.6 Ab wann lohnt es sich?","anker":"sec:metaheuristiken-umschlagpunkt","text":"„Kleine Probleme exakt, große mit Metaheuristiken“ ist richtig und nutzlos, solange niemand sagt, wo groß anfängt. Diese Zahl lässt sich messen."},{"seite":"metaheuristiken.html","titel":"Die Spalte, die man nicht weglassen darf","anker":"die-spalte-die-man-nicht-weglassen-darf","text":"Bei 500 Aufträgen haben wir die CP-SAT-Lösung verworfen. Seine Schranke ist trotzdem das Wertvollste in der ganzen Tabelle."},{"seite":"metaheuristiken.html","titel":"9.7 Large Neighborhood Search","anker":"sec:metaheuristiken-lns","text":"Die Tabelle oben endet mit einem Widerspruch. Bei 500 Aufträgen scheitert der exakte Solver an der Größe; die Metaheuristik scheitert an der Kleinheit ihrer Züge — sie verschiebt j"},{"seite":"metaheuristiken.html","titel":"Das Fenster ist die eine Stellschraube","anker":"das-fenster-ist-die-eine-stellschraube","text":""},{"seite":"metaheuristiken.html","titel":"Das Gesamtbild","anker":"das-gesamtbild","text":""},{"seite":"metaheuristiken.html","titel":"9.8 Was dieses Kapitel nicht behandelt","anker":"sec:metaheuristiken-abgrenzung","text":"Damit die Landkarte vollständig ist — drei Verfahren, die in dieselbe Familie gehören und hier bewusst nur benannt werden:"},{"seite":"metaheuristiken.html","titel":"9.9 Übungsaufgaben","anker":"sec:metaheuristiken-uebungsaufgaben","text":""},{"seite":"metaheuristiken.html","titel":"9.10 Finde den Denkfehler","anker":"sec:metaheuristiken-denkfehler","text":""},{"seite":"metaheuristiken.html","titel":"9.11 Micro-Quiz","anker":"sec:metaheuristiken-quiz","text":""},{"seite":"metaheuristiken.html","titel":"9.12 Selbsttest","anker":"sec:metaheuristiken-selbsttest","text":""},{"seite":"metaheuristiken.html","titel":"9.13 Zusammenfassung","anker":"sec:metaheuristiken-zusammenfassung","text":""},{"seite":"dekomposition.html","titel":"Kapitel 10: Spaltengenerierung — das Modell umbauen statt die Lösung raten","anker":null,"text":""},{"seite":"dekomposition.html","titel":"10.1 In 5 Minuten gelöst","anker":"sec:dekomposition-schnellstart","text":""},{"seite":"dekomposition.html","titel":"10.2 Lernziele","anker":"sec:dekomposition-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"dekomposition.html","titel":"10.3 Warum das naheliegende Modell scheitert","anker":"sec:dekomposition-symmetrie","text":"Der erste Entwurf eines Zuschnittmodells sieht fast immer so aus: eine Binärvariable z_{jr} für „Stück j liegt auf Rolle r“, dazu y_r für „Rolle r wird benutzt”."},{"seite":"dekomposition.html","titel":"10.4 Rechnen mit Mustern, ohne sie aufzuschreiben","anker":"sec:dekomposition-schleife","text":"Die Idee ist alt (Gilmore und Gomory, 1961) und in einem Satz gesagt:"},{"seite":"dekomposition.html","titel":"10.5 Das Programm","anker":"sec:dekomposition-programm","text":""},{"seite":"dekomposition.html","titel":"10.6 Was die Zahlen zeigen","anker":"sec:dekomposition-befund","text":""},{"seite":"dekomposition.html","titel":"Die Musterzahl explodiert, der Bedarf nicht","anker":"die-musterzahl-explodiert-der-bedarf-nicht","text":""},{"seite":"dekomposition.html","titel":"Der unbequeme Teil: Es lohnt sich nicht immer","anker":"der-unbequeme-teil-es-lohnt-sich-nicht-immer","text":""},{"seite":"dekomposition.html","titel":"10.7 Übungsaufgaben","anker":"sec:dekomposition-uebungsaufgaben","text":""},{"seite":"dekomposition.html","titel":"10.8 Finde den Denkfehler","anker":"sec:dekomposition-denkfehler","text":""},{"seite":"dekomposition.html","titel":"10.9 Micro-Quiz","anker":"sec:dekomposition-quiz","text":""},{"seite":"dekomposition.html","titel":"10.10 Selbsttest","anker":"sec:dekomposition-selbsttest","text":""},{"seite":"dekomposition.html","titel":"10.11 Zusammenfassung","anker":"sec:dekomposition-zusammenfassung","text":""},{"seite":"synthese-kernverfahren.html","titel":"Synthese Teil II","anker":null,"text":""},{"seite":"synthese-kernverfahren.html","titel":"Die Entscheidungsmatrix","anker":"die-entscheidungsmatrix","text":""},{"seite":"synthese-kernverfahren.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-kernverfahren.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-kernverfahren.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"qp-nlp.html","titel":"Kapitel 11: Quadratische und nichtlineare Optimierung — KKT, Lagrange, Konvexität","anker":null,"text":""},{"seite":"qp-nlp.html","titel":"11.1 In 5 Minuten gelöst","anker":"sec:qp-nlp-schnellstart","text":""},{"seite":"qp-nlp.html","titel":"11.2 Lernziele","anker":"sec:qp-nlp-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"qp-nlp.html","titel":"Auffrischung: Gradient in einer Minute","anker":"auffrischung-gradient-in-einer-minute","text":"Der Gradient \\nabla f(\\mathbf{x}) ist der Vektor aller partiellen Ableitungen:"},{"seite":"qp-nlp.html","titel":"11.3 Das quadratische Programm (QP)","anker":"sec:qp-nlp-das-quadratische-programm-qp","text":"Ein quadratisches Programm hat eine quadratische Zielfunktion und lineare Nebenbedingungen:"},{"seite":"qp-nlp.html","titel":"Konvexität: die präzise Aussage","anker":"konvexität-die-präzise-aussage","text":"Eine verbreitete, aber zu stark formulierte Aussage lautet: „Wenn \\mathbf{P} positiv semi-definit ist, ist das QP streng konvex; es existiert ein eindeutiges globales Minimum.“ Kor"},{"seite":"qp-nlp.html","titel":"11.4 Die Karush-Kuhn-Tucker-Bedingungen","anker":"sec:qp-nlp-die-karush-kuhn-tucker-bedingungen","text":"Die KKT-Bedingungen verallgemeinern den Lagrange-Ansatz auf Probleme mit Ungleichungen. Sie sind notwendig für ein Optimum — und bei konvexen Problemen auch hinreichend."},{"seite":"qp-nlp.html","titel":"Die vier KKT-Bedingungen","anker":"die-vier-kkt-bedingungen","text":"Am Optimum (\\mathbf{x}^*, \\boldsymbol{\\lambda}^*, \\boldsymbol{\\nu}^*) gilt:"},{"seite":"qp-nlp.html","titel":"11.5 Nichtlineare Optimierung mit scipy.optimize.minimize","anker":"sec:qp-nlp-nichtlineare-optimierung-mit-scipy-optimize-minimize","text":"Wenn Zielfunktion oder Nebenbedingungen weder linear noch quadratisch sind — Logarithmen, Wurzeln, Exponentialfunktionen — nutzt man gradientenbasierte Verfahren wie SLSQP (Sequent"},{"seite":"qp-nlp.html","titel":"Praxisfall: Entropie-maximierte Kapitalallokation","anker":"praxisfall-entropie-maximierte-kapitalallokation","text":"Wir optimieren ein Portfolio nicht nur nach Varianz, sondern maximieren zusätzlich die Shannon-Entropie der Gewichte, um Klumpenrisiken glatt zu bestrafen:"},{"seite":"qp-nlp.html","titel":"11.6 Jenseits der Konvexität: lokale Optima, Multistart und MINLP","anker":"sec:qp-nlp-nichtkonvex","text":"Alles bisher in diesem Kapitel stand unter einer Bedingung: Konvexität. Sie ist der Grund, warum CVXPY eine Optimalitätsgarantie geben kann und warum die KKT-Bedingungen nicht nur "},{"seite":"qp-nlp.html","titel":"Woher Nicht-Konvexität im Alltag kommt","anker":"woher-nicht-konvexität-im-alltag-kommt","text":"Nicht aus exotischer Mathematik, sondern aus ganz gewöhnlichen betriebswirtschaftlichen Effekten:"},{"seite":"qp-nlp.html","titel":"Was ein lokales Optimum praktisch bedeutet","anker":"was-ein-lokales-optimum-praktisch-bedeutet","text":"scipy.optimize.minimize verweigert nichts. Es rechnet, meldet success: True und liefert eine Zahl. Diese Meldung heißt aber nicht „das ist das Optimum“, sondern nur:"},{"seite":"qp-nlp.html","titel":"Die Zahlen im Klartext","anker":"die-zahlen-im-klartext","text":""},{"seite":"qp-nlp.html","titel":"Ausblick: MINLP und globale Solver","anker":"ausblick-minlp-und-globale-solver","text":"Wird zusätzlich noch ganzzahlig entschieden — „welche Anlage läuft überhaupt“ und „wie stark“ —, entsteht ein MINLP (Mixed-Integer Nonlinear Program), die schwierigste der in diese"},{"seite":"qp-nlp.html","titel":"11.7 Übungsaufgaben","anker":"sec:qp-nlp-uebungsaufgaben","text":""},{"seite":"qp-nlp.html","titel":"11.8 Finde den Denkfehler","anker":"sec:qp-nlp-denkfehler","text":""},{"seite":"qp-nlp.html","titel":"11.9 Micro-Quiz","anker":"sec:qp-nlp-quiz","text":""},{"seite":"qp-nlp.html","titel":"11.10 Selbsttest","anker":"sec:qp-nlp-selbsttest","text":""},{"seite":"qp-nlp.html","titel":"11.11 Zusammenfassung","anker":"sec:qp-nlp-zusammenfassung","text":""},{"seite":"unsicherheit.html","titel":"Kapitel 12: Optimierung unter Unsicherheit — Monte-Carlo, Stochastik, Robustheit","anker":null,"text":""},{"seite":"unsicherheit.html","titel":"12.1 In 5 Minuten gelöst","anker":"sec:unsicherheit-schnellstart","text":""},{"seite":"unsicherheit.html","titel":"12.2 Lernziele","anker":"sec:unsicherheit-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"unsicherheit.html","titel":"12.3 Der Fluch des Durchschnitts","anker":"sec:unsicherheit-der-fluch-des-durchschnitts","text":"Der naheliegende Umgang mit Unsicherheit lautet: „Wir setzen den Erwartungswert ein und rechnen deterministisch.“ Das ist fast immer falsch — und zwar nicht ungenau, sondern system"},{"seite":"unsicherheit.html","titel":"Die vier Ansätze im Überblick","anker":"die-vier-ansätze-im-überblick","text":""},{"seite":"unsicherheit.html","titel":"12.4 Monte-Carlo-Simulation","anker":"sec:unsicherheit-monte-carlo-simulation","text":"Monte-Carlo heißt schlicht: Ziehe sehr viele Zufallsszenarien, werte für jedes den Plan aus, und betrachte die Verteilung der Ergebnisse statt eines einzigen Werts."},{"seite":"unsicherheit.html","titel":"12.5 Zweistufige stochastische Programmierung","anker":"sec:unsicherheit-zweistufige-stochastische-programmierung","text":"Das zweistufige Modell mit Recourse bildet ab, dass Entscheidungen zeitlich gestaffelt sind:"},{"seite":"unsicherheit.html","titel":"12.6 Robuste Optimierung: gegen den Worst Case absichern","anker":"sec:unsicherheit-robuste-optimierung-gegen-den-worst-case","text":"Wenn Wahrscheinlichkeiten unbekannt oder instabil sind — Marktcrashs, Lieferkettenabrisse, Pandemien —, hilft die robuste Optimierung. Sie fragt nicht nach dem Mittel, sondern nach"},{"seite":"unsicherheit.html","titel":"12.7 Wahrscheinlichkeitsbeschränkungen: „mit 95 % Sicherheit“","anker":"sec:unsicherheit-chance-constraints","text":"Die drei bisherigen Ansätze beantworten Fragen, die im Sitzungssaal selten so gestellt werden. „Was ist im Mittel am besten?“ hört ein Vorstand als Ausflucht; „was hält auch im sch"},{"seite":"unsicherheit.html","titel":"Weg 1: Normalverteilung — die Bedingung wird ein Kegel","anker":"weg-1-normalverteilung-die-bedingung-wird-ein-kegel","text":"Ist \\mathbf{a} \\sim \\mathcal{N}(\\hat{\\mathbf{a}}, \\boldsymbol{\\Sigma}), dann ist \\mathbf{a}^\\top\\mathbf{x} für festes \\mathbf{x} selbst normalverteilt, mit Erwartungswert \\hat{\\mat"},{"seite":"unsicherheit.html","titel":"Weg 2: Szenarien — Big-M ohne Verteilungsannahme","anker":"weg-2-szenarien-big-m-ohne-verteilungsannahme","text":"Liegen statt einer Verteilung nur S beobachtete Szenarien vor, bekommt jedes eine Binärvariable z_s, die sagt, ob es verletzt werden darf:"},{"seite":"unsicherheit.html","titel":"Der Fall: ein Kraftwerkspark mit 500 MW Zusage","anker":"der-fall-ein-kraftwerkspark-mit-500-mw-zusage","text":""},{"seite":"unsicherheit.html","titel":"12.8 Übungsaufgaben","anker":"sec:unsicherheit-uebungsaufgaben","text":""},{"seite":"unsicherheit.html","titel":"12.9 Finde den Denkfehler","anker":"sec:unsicherheit-denkfehler","text":""},{"seite":"unsicherheit.html","titel":"12.10 Micro-Quiz","anker":"sec:unsicherheit-quiz","text":""},{"seite":"unsicherheit.html","titel":"12.11 Selbsttest","anker":"sec:unsicherheit-selbsttest","text":""},{"seite":"unsicherheit.html","titel":"12.12 Zusammenfassung","anker":"sec:unsicherheit-zusammenfassung","text":""},{"seite":"dynamische-programmierung.html","titel":"Kapitel 13: Dynamische Programmierung — Die Bellman-Gleichung und Order-Execution","anker":null,"text":""},{"seite":"dynamische-programmierung.html","titel":"13.1 In 5 Minuten gelöst","anker":"sec:dynamische-programmierung-schnellstart","text":""},{"seite":"dynamische-programmierung.html","titel":"13.2 Lernziele","anker":"sec:dynamische-programmierung-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"dynamische-programmierung.html","titel":"13.3 Das Bellmansche Optimalitätsprinzip","anker":"sec:dynamische-programmierung-das-bellmansche-optimalitaetsprinzip","text":"Richard Bellman formulierte 1957 das Grundprinzip der Dynamischen Programmierung (DP):"},{"seite":"dynamische-programmierung.html","titel":"Die vier Bausteine eines DP-Modells","anker":"die-vier-bausteine-eines-dp-modells","text":""},{"seite":"dynamische-programmierung.html","titel":"Die Bellman-Gleichung","anker":"die-bellman-gleichung","text":"V_t(S_t) = \\min_{a_t\\in\\mathcal{A}(S_t)}\\Big\\{\\,C(S_t,a_t) \\;+\\; \\gamma\\cdot\\mathbb{E}\\big[V_{t+1}(S_{t+1})\\mid S_t,a_t\\big]\\Big\\}"},{"seite":"dynamische-programmierung.html","titel":"13.4 Das Almgren-Chriss-Problem: optimale Orderausführung","anker":"sec:dynamische-programmierung-das-almgren-chriss-problem-optimale-orderausfuehrung","text":"Ein klassisches OR-Problem an Börsen: Ein institutioneller Händler muss X_0 = 100\\,000 Aktien innerhalb von T Handelsperioden verkaufen."},{"seite":"dynamische-programmierung.html","titel":"13.5 Der Fluch der Dimensionalität","anker":"sec:dynamische-programmierung-der-fluch-der-dimensionalitaet","text":"DP ist mächtig, aber es hat eine harte Grenze. Der Aufwand der Rückwärtsinduktion ist"},{"seite":"dynamische-programmierung.html","titel":"13.6 Übungsaufgaben","anker":"sec:dynamische-programmierung-uebungsaufgaben","text":""},{"seite":"dynamische-programmierung.html","titel":"13.7 Finde den Denkfehler","anker":"sec:dynamische-programmierung-denkfehler","text":"Bei dynamischer Programmierung entscheidet eine einzige Frage über Erfolg oder Misserfolg: Was gehört in den Zustand? Ist er zu klein, rechnet das Verfahren völlig korrekt — nur eb"},{"seite":"dynamische-programmierung.html","titel":"13.8 Micro-Quiz","anker":"sec:dynamische-programmierung-quiz","text":""},{"seite":"dynamische-programmierung.html","titel":"13.9 Selbsttest","anker":"sec:dynamische-programmierung-selbsttest","text":""},{"seite":"dynamische-programmierung.html","titel":"13.10 Zusammenfassung","anker":"sec:dynamische-programmierung-zusammenfassung","text":""},{"seite":"mehrziel.html","titel":"Kapitel 14: Mehrere Ziele — Pareto-Fronten statt Gewichte","anker":null,"text":""},{"seite":"mehrziel.html","titel":"14.1 In 5 Minuten gelöst","anker":"sec:mehrziel-schnellstart","text":""},{"seite":"mehrziel.html","titel":"14.2 Lernziele","anker":"sec:mehrziel-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"mehrziel.html","titel":"14.3 Wann ein Kompromiss gut ist","anker":"sec:mehrziel-dominanz","text":"Bei einem Ziel ist „besser” eindeutig. Bei zwei Zielen nicht mehr: Ein Plan kann billiger und schmutziger sein als ein anderer, und dann ist keiner von beiden besser."},{"seite":"mehrziel.html","titel":"14.4 Der Reflex: alles in ein Ziel rühren","anker":"sec:mehrziel-skalarisierung","text":"Der naheliegende Weg, zwei Ziele zu einem zu machen, ist die gewichtete Summe:"},{"seite":"mehrziel.html","titel":"14.5 Das Programm","anker":"sec:mehrziel-programm","text":"Zwölf Sendungen, drei Verkehrsträger, fünf freie Bahntrassen. Der Zielkonflikt entsteht nicht zwischen den Trägern — die Bahn ist immer billiger und sauberer als der LKW —, sondern"},{"seite":"mehrziel.html","titel":"14.6 Was die gewichtete Summe nicht sieht","anker":"sec:mehrziel-luecke","text":"Die Front hat zehn Punkte. Ein Raster aus 1 401 Gewichten findet sechs davon. Vier Kompromisse sind für kein Gewicht erreichbar — und das ist keine Frage der Rasterweite."},{"seite":"mehrziel.html","titel":"Das Gewicht ist keine Feineinstellung","anker":"das-gewicht-ist-keine-feineinstellung","text":"Die zweite Beobachtung aus derselben Tabelle: 1 401 Gewichte erzeugen sechs verschiedene Pläne. Weite Gewichtsbereiche liefern dasselbe Ergebnis, und dazwischen springt es."},{"seite":"mehrziel.html","titel":"14.7 Das ε-Constraint-Verfahren","anker":"sec:mehrziel-epsilon","text":"Der Ausweg ist unspektakulär: Man macht aus dem zweiten Ziel eine Nebenbedingung."},{"seite":"mehrziel.html","titel":"Wenn es eine klare Rangfolge gibt: lexikografisch","anker":"wenn-es-eine-klare-rangfolge-gibt-lexikografisch","text":"Manchmal ist ein Ziel wirklich vorrangig. Dann optimiert man lexikografisch: erst das erste Ziel, dann das zweite unter der Bedingung, dass das erste (fast) erhalten bleibt."},{"seite":"mehrziel.html","titel":"14.8 Wie man die Front vorlegt","anker":"sec:mehrziel-vorlegen","text":"Eine Front mit zehn Punkten ist kein Ergebnis, sondern eine Speisekarte. Damit sie entscheidbar wird, gehört eine Spalte dazu, die im Programm EUR je kg heißt:"},{"seite":"mehrziel.html","titel":"14.9 Übungsaufgaben","anker":"sec:mehrziel-uebungsaufgaben","text":""},{"seite":"mehrziel.html","titel":"14.10 Finde den Denkfehler","anker":"sec:mehrziel-denkfehler","text":""},{"seite":"mehrziel.html","titel":"14.11 Micro-Quiz","anker":"sec:mehrziel-quiz","text":""},{"seite":"mehrziel.html","titel":"14.12 Selbsttest","anker":"sec:mehrziel-selbsttest","text":""},{"seite":"mehrziel.html","titel":"14.13 Zusammenfassung","anker":"sec:mehrziel-zusammenfassung","text":""},{"seite":"prognose.html","titel":"Kapitel 15: Predict-then-Optimize — die bessere Prognose, die schlechtere Entscheidung","anker":null,"text":""},{"seite":"prognose.html","titel":"15.1 In 5 Minuten gelöst","anker":"sec:prognose-schnellstart","text":""},{"seite":"prognose.html","titel":"15.2 Lernziele","anker":"sec:prognose-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"prognose.html","titel":"15.3 Die Naht zwischen zwei Abteilungen","anker":"sec:prognose-naht","text":"In fast jedem Unternehmen sieht der Ablauf so aus:"},{"seite":"prognose.html","titel":"15.4 Das Programm","anker":"sec:prognose-programm","text":"Dieselbe Bäckerei, aber mit Merkmalen: Wochentag, Temperatur, Aktionstage. Die Nachfrage muss jetzt modelliert werden."},{"seite":"prognose.html","titel":"Was Sie in der Abbildung sehen. Links und rechts gewinnt nicht dasselbe Verfahren. Die Punktprognose hat den besten Prognosefehler und die höchsten Kosten — beide Maßstäbe messen etwas anderes, und nur einer davon steht am Ende in der Bilanz.","anker":"was-sie-in-der-abbildung-sehen.-links-und-rechts-gewinnt-nicht-dasselbe-verfahren.-die-punktprognose-hat-den-besten-prognosefehler-und-die-höchsten-kosten-beide-maßstäbe-messen-etwas-anderes-und-nur-einer-davon-steht-am-ende-in-der-bilanz.","text":""},{"seite":"prognose.html","titel":"15.5 Der Befund","anker":"sec:prognose-befund","text":""},{"seite":"prognose.html","titel":"Warum ein pauschaler Zuschlag zu kurz greift","anker":"warum-ein-pauschaler-zuschlag-zu-kurz-greift","text":"Die zweite Zeile der Tabelle ist die verbreitete Praxis: Punktprognose plus Sicherheitsbestand. Sie hilft (−7,3 %), bleibt aber hinter der Quantilregression zurück. Das Programm re"},{"seite":"prognose.html","titel":"15.6 Eine Messfalle, in die der Autor selbst getappt ist","anker":"sec:prognose-messfalle","text":"Der erste Entwurf dieses Programms bewertete auf 230 Testtagen — ein realistischer Zeitraum, gut sieben Monate. Dort hatte die Quantilregression den besseren MSE, und die ganze Aus"},{"seite":"prognose.html","titel":"15.7 Wie weit das Verfahren reicht","anker":"sec:prognose-ausblick","text":"Die dritte Zeile der Ergebnistabelle — „Zuschlag auf Kosten trainiert” — ist die einfachste denkbare Form einer Idee, die derzeit erforscht wird: das Prognosemodell direkt auf die "},{"seite":"prognose.html","titel":"15.8 Übungsaufgaben","anker":"sec:prognose-uebungsaufgaben","text":""},{"seite":"prognose.html","titel":"15.9 Finde den Denkfehler","anker":"sec:prognose-denkfehler","text":""},{"seite":"prognose.html","titel":"15.10 Micro-Quiz","anker":"sec:prognose-quiz","text":""},{"seite":"prognose.html","titel":"15.11 Selbsttest","anker":"sec:prognose-selbsttest","text":""},{"seite":"prognose.html","titel":"15.12 Zusammenfassung","anker":"sec:prognose-zusammenfassung","text":""},{"seite":"synthese-nichtlinear.html","titel":"Synthese Teil III","anker":null,"text":""},{"seite":"synthese-nichtlinear.html","titel":"Welche Annahme fällt weg?","anker":"welche-annahme-fällt-weg","text":""},{"seite":"synthese-nichtlinear.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-nichtlinear.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-nichtlinear.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"bruecke.html","titel":"Kapitel 16: Die Strukturbrücke — dieselbe Mathematik, zwei Welten","anker":null,"text":""},{"seite":"bruecke.html","titel":"16.1 In 5 Minuten gelöst","anker":"sec:bruecke-schnellstart","text":""},{"seite":"bruecke.html","titel":"16.2 Lernziele","anker":"sec:bruecke-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"bruecke.html","titel":"16.3 Das Programm","anker":"sec:bruecke-programm","text":""},{"seite":"bruecke.html","titel":"16.4 Die zwei Brücken im Einzelnen","anker":"sec:bruecke-einzelnen","text":""},{"seite":"bruecke.html","titel":"Erste Brücke: Allokation","anker":"erste-brücke-allokation","text":""},{"seite":"bruecke.html","titel":"Zweite Brücke: Absicherung gegen den schlechtesten Fall","anker":"zweite-brücke-absicherung-gegen-den-schlechtesten-fall","text":"Hier ist die Übereinstimmung noch enger, weil sie nicht nur die Struktur betrifft, sondern die Formel. Die Rockafellar-Uryasev-Formulierung aus Kapitel 20"},{"seite":"bruecke.html","titel":"Wo die Lösungen sich unterscheiden","anker":"wo-die-lösungen-sich-unterscheiden","text":"Der Vergleich fördert nebenbei einen Unterschied zutage, den man nicht erwartet: Die Streuungsgrenze von 40 % bindet nur im Einkauf, nicht im Depot."},{"seite":"bruecke.html","titel":"16.5 Wo die Brücke endet","anker":"sec:bruecke-grenzen","text":"Eine Analogie ist nur so nützlich wie die Grenzen, die man ihr zieht. Drei Dinge wandern nicht mit:"},{"seite":"bruecke.html","titel":"16.6 Übungsaufgaben","anker":"sec:bruecke-uebungsaufgaben","text":""},{"seite":"bruecke.html","titel":"16.7 Finde den Denkfehler","anker":"sec:bruecke-denkfehler","text":""},{"seite":"bruecke.html","titel":"16.8 Micro-Quiz","anker":"sec:bruecke-quiz","text":""},{"seite":"bruecke.html","titel":"16.9 Selbsttest","anker":"sec:bruecke-selbsttest","text":""},{"seite":"bruecke.html","titel":"16.10 Zusammenfassung","anker":"sec:bruecke-zusammenfassung","text":""},{"seite":"supplychain.html","titel":"Kapitel 17: Supply-Chain und Energieeinsatz unter Unsicherheit","anker":null,"text":""},{"seite":"supplychain.html","titel":"17.1 In 5 Minuten gelöst","anker":"sec:supplychain-schnellstart","text":""},{"seite":"supplychain.html","titel":"17.2 Lernziele","anker":"sec:supplychain-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"supplychain.html","titel":"17.3 Die Aufgabe","anker":"sec:supplychain-aufgabe","text":"Fünf Blöcke, 24 Stunden, eine Last zwischen 270 und 804 MW. Dazu Windeinspeisung, die niemand am Vorabend kennt."},{"seite":"supplychain.html","titel":"Die Zeitstruktur der Entscheidung","anker":"die-zeitstruktur-der-entscheidung","text":"Hier geht das Kapitel über Abschnitt 12.5 hinaus:"},{"seite":"supplychain.html","titel":"17.4 Das Programm","anker":"sec:supplychain-programm","text":""},{"seite":"supplychain.html","titel":"17.5 Der Befund","anker":"sec:supplychain-befund","text":""},{"seite":"supplychain.html","titel":"17.6 Was Versorgungssicherheit kostet","anker":"sec:supplychain-sicherheit","text":"Der dritte Plan legt eine CVaR-Schranke auf die nicht gedeckte Energie: Im Mittel der schlechtesten 10 % der Szenarien darf nichts fehlen. Dieselbe Rockafellar-Uryasev-Konstruktion"},{"seite":"supplychain.html","titel":"17.7 Übungsaufgaben","anker":"sec:supplychain-uebungsaufgaben","text":""},{"seite":"supplychain.html","titel":"17.8 Finde den Denkfehler","anker":"sec:supplychain-denkfehler","text":""},{"seite":"supplychain.html","titel":"17.9 Micro-Quiz","anker":"sec:supplychain-quiz","text":""},{"seite":"supplychain.html","titel":"17.10 Selbsttest","anker":"sec:supplychain-selbsttest","text":""},{"seite":"supplychain.html","titel":"17.11 Zusammenfassung","anker":"sec:supplychain-zusammenfassung","text":""},{"seite":"finanzdaten.html","titel":"Kapitel 18: Finanzdaten-Modellierung — Renditen, Kovarianz und Shrinkage","anker":null,"text":""},{"seite":"finanzdaten.html","titel":"18.1 In 5 Minuten gelöst","anker":"sec:finanzdaten-schnellstart","text":""},{"seite":"finanzdaten.html","titel":"18.2 Lernziele","anker":"sec:finanzdaten-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"finanzdaten.html","titel":"18.3 Diskrete und logarithmische Renditen","anker":"sec:finanzdaten-diskrete-und-logarithmische-renditen","text":"Sei P_{i,t} der bereinigte Schlusskurs (adjusted close) von Titel i zum Zeitpunkt t. „Bereinigt“ heißt: um Dividenden und Aktiensplits korrigiert — sonst erscheint jede Dividendenz"},{"seite":"finanzdaten.html","titel":"Diskrete Rendite","anker":"diskrete-rendite","text":"R_{i,t} = \\frac{P_{i,t} - P_{i,t-1}}{P_{i,t-1}} = \\frac{P_{i,t}}{P_{i,t-1}} - 1"},{"seite":"finanzdaten.html","titel":"Logarithmische Rendite","anker":"logarithmische-rendite","text":"r_{i,t} = \\ln\\!\\left(\\frac{P_{i,t}}{P_{i,t-1}}\\right) = \\ln P_{i,t} - \\ln P_{i,t-1}"},{"seite":"finanzdaten.html","titel":"18.4 Das Schätzfehler-Problem","anker":"sec:finanzdaten-das-schaetzfehler-problem","text":"Sei \\mathbf{X}\\in\\mathbb{R}^{T\\times N} die Matrix zentrierter Renditen von N Titeln über T Handelstage. Die Stichproben-Kovarianzmatrix lautet:"},{"seite":"finanzdaten.html","titel":"18.5 Ledoit-Wolf-Shrinkage","anker":"sec:finanzdaten-ledoit-wolf-shrinkage","text":"Olivier Ledoit und Michael Wolf lösten das Dilemma durch lineare Schrumpfung:"},{"seite":"finanzdaten.html","titel":"Welches Ziel \\mathbf{F}?","anker":"welches-ziel-mathbff","text":"Ein häufiges Missverständnis: Es gibt nämlich mehrere Ledoit-Wolf-Varianten, und Text und Code müssen sich konsistent auf dieselbe beziehen:"},{"seite":"finanzdaten.html","titel":"18.6 Praxis: Datenpipeline mit korrekter Spaltenreihenfolge","anker":"sec:finanzdaten-praxis-datenpipeline-mit-korrekter-spaltenreihenfolge","text":""},{"seite":"finanzdaten.html","titel":"18.7 Übungsaufgaben","anker":"sec:finanzdaten-uebungsaufgaben","text":""},{"seite":"finanzdaten.html","titel":"18.8 Finde den Denkfehler","anker":"sec:finanzdaten-denkfehler","text":""},{"seite":"finanzdaten.html","titel":"18.9 Micro-Quiz","anker":"sec:finanzdaten-quiz","text":""},{"seite":"finanzdaten.html","titel":"18.10 Selbsttest","anker":"sec:finanzdaten-selbsttest","text":""},{"seite":"finanzdaten.html","titel":"18.11 Zusammenfassung","anker":"sec:finanzdaten-zusammenfassung","text":""},{"seite":"markowitz.html","titel":"Kapitel 19: Die moderne Portfoliotheorie nach Markowitz","anker":null,"text":""},{"seite":"markowitz.html","titel":"19.1 In 5 Minuten gelöst","anker":"sec:markowitz-schnellstart","text":""},{"seite":"markowitz.html","titel":"19.2 Lernziele","anker":"sec:markowitz-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"markowitz.html","titel":"19.3 Warum Diversifikation funktioniert","anker":"sec:markowitz-warum-diversifikation-funktioniert","text":"Harry Markowitz’ zentrale Einsicht von 1952: Das Risiko eines Portfolios ist nicht die Summe der Einzelrisiken. Es hängt davon ab, wie die Titel zusammenspielen."},{"seite":"markowitz.html","titel":"19.4 Das Mean-Variance-Modell","anker":"sec:markowitz-das-mean-variance-modell","text":"Sei \\mathbf{w} = (w_1,\\dots,w_n)^\\top der Vektor der Portfoliogewichte:"},{"seite":"markowitz.html","titel":"Die Effizienzgrenze und das Tangentialportfolio","anker":"die-effizienzgrenze-und-das-tangentialportfolio","text":"Die Effizienzgrenze (efficient frontier) ist die Kurve aller Portfolios, die zu jedem Risikoniveau die höchstmögliche Rendite liefern."},{"seite":"markowitz.html","titel":"Die Korn-Transformation","anker":"die-korn-transformation","text":"Die Sharpe Ratio ist ein Bruch — und damit nicht konvex. CVXPY würde sie ablehnen. Der Ausweg ist ein eleganter Trick, der die Homogenität des Problems ausnutzt:"},{"seite":"markowitz.html","titel":"19.5 Institutionelle Nebenbedingungen","anker":"sec:markowitz-institutionelle-nebenbedingungen","text":"In der Praxis gelten regulatorische und interne Regeln:"},{"seite":"markowitz.html","titel":"Was diese Regeln kosten","anker":"was-diese-regeln-kosten","text":"Jede Regel schneidet etwas vom Erreichbaren ab. Wie viel, lässt sich ausrechnen — man legt die Effizienzlinie ohne und mit Beschränkungen übereinander:"},{"seite":"markowitz.html","titel":"19.6 Vollimplementierung mit CVXPY","anker":"sec:markowitz-vollimplementierung-mit-cvxpy","text":""},{"seite":"markowitz.html","titel":"19.7 Übungsaufgaben","anker":"sec:markowitz-uebungsaufgaben","text":""},{"seite":"markowitz.html","titel":"19.8 Finde den Denkfehler","anker":"sec:markowitz-denkfehler","text":"Kapitel 18 hat gezeigt, was eine schlecht geschätzte Kovarianzmatrix anrichtet. Bei den erwarteten Renditen ist es schlimmer — und weil sie in der Zielfunktion stehen statt in der "},{"seite":"markowitz.html","titel":"19.9 Micro-Quiz","anker":"sec:markowitz-quiz","text":""},{"seite":"markowitz.html","titel":"19.10 Selbsttest","anker":"sec:markowitz-selbsttest","text":""},{"seite":"markowitz.html","titel":"19.11 Zusammenfassung","anker":"sec:markowitz-zusammenfassung","text":""},{"seite":"cvar.html","titel":"Kapitel 20: Tail-Risiko, CVaR und Transaktionskosten","anker":null,"text":""},{"seite":"cvar.html","titel":"20.1 In 5 Minuten gelöst","anker":"sec:cvar-schnellstart","text":""},{"seite":"cvar.html","titel":"20.2 Lernziele","anker":"sec:cvar-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"cvar.html","titel":"20.3 Die zwei Schwächen des Markowitz-Modells","anker":"sec:cvar-die-zwei-schwaechen-des-markowitz-modells","text":"Erstens: die Normalverteilungs-Illusion. Die Varianz behandelt Aufwärts- und Abwärtsschwankungen gleich und unterstellt implizit symmetrische, dünn auslaufende Verteilungen. Reale "},{"seite":"cvar.html","titel":"20.4 Value at Risk und Conditional Value at Risk","anker":"sec:cvar-value-at-risk-und-conditional-value","text":""},{"seite":"cvar.html","titel":"Das Rockafellar-Uryasev-Theorem (2000)","anker":"das-rockafellar-uryasev-theorem-2000","text":"Rockafellar und Uryasev zeigten, dass sich der CVaR über S Szenarien exakt als lineares Programm formulieren lässt:"},{"seite":"cvar.html","titel":"20.5 Transaktionskosten über die L_1-Norm","anker":"sec:cvar-transaktionskosten-ueber-die-l-1-norm","text":"Sei \\mathbf{w}_{\\text{alt}} das bestehende Portfolio und \\mathbf{w} das neue Ziel. Der Umschlag (turnover) ist:"},{"seite":"cvar.html","titel":"20.6 Implementierung: CVaR-Portfolio mit Reibung","anker":"sec:cvar-implementierung-cvar-portfolio-mit-reibung","text":""},{"seite":"cvar.html","titel":"20.7 Übungsaufgaben","anker":"sec:cvar-uebungsaufgaben","text":""},{"seite":"cvar.html","titel":"20.8 Finde den Denkfehler","anker":"sec:cvar-denkfehler","text":""},{"seite":"cvar.html","titel":"20.9 Micro-Quiz","anker":"sec:cvar-quiz","text":""},{"seite":"cvar.html","titel":"20.10 Selbsttest","anker":"sec:cvar-selbsttest","text":""},{"seite":"cvar.html","titel":"20.11 Zusammenfassung","anker":"sec:cvar-zusammenfassung","text":""},{"seite":"handelsmaschine.html","titel":"Kapitel 21: Die vollständige quantitative Handelsmaschine","anker":null,"text":""},{"seite":"handelsmaschine.html","titel":"21.1 In 5 Minuten gelöst","anker":"sec:handelsmaschine-schnellstart","text":""},{"seite":"handelsmaschine.html","titel":"21.2 Lernziele","anker":"sec:handelsmaschine-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"handelsmaschine.html","titel":"21.3 Die Architektur","anker":"sec:handelsmaschine-die-architektur","text":""},{"seite":"handelsmaschine.html","titel":"21.4 Rebalancing-Termine richtig bestimmen","anker":"sec:handelsmaschine-rebalancing-termine-richtig-bestimmen","text":""},{"seite":"handelsmaschine.html","titel":"21.5 Die Engine","anker":"sec:handelsmaschine-die-engine","text":""},{"seite":"handelsmaschine.html","titel":"21.6 Die fünf Selbsttäuschungen des Backtestens","anker":"sec:handelsmaschine-die-fuenf-selbsttaeuschungen-des-backtestens","text":""},{"seite":"handelsmaschine.html","titel":"21.7 Vom Backtest zum Betrieb","anker":"sec:handelsmaschine-vom-backtest-zum-betrieb","text":"Ein guter Backtest ist eine notwendige, keine hinreichende Bedingung. Was zwischen Backtest und echtem Einsatz liegt:"},{"seite":"handelsmaschine.html","titel":"21.8 Übungsaufgaben","anker":"sec:handelsmaschine-uebungsaufgaben","text":""},{"seite":"handelsmaschine.html","titel":"21.9 Finde den Denkfehler","anker":"sec:handelsmaschine-denkfehler","text":"Backtest_Fallen.py zeigt drei Fehler, die im Backtest stecken: Lookahead, Survivorship, vergessene Kosten. Alle drei kann man einem Programm ansehen, wenn man es liest."},{"seite":"handelsmaschine.html","titel":"21.10 Micro-Quiz","anker":"sec:handelsmaschine-quiz","text":""},{"seite":"handelsmaschine.html","titel":"21.11 Selbsttest","anker":"sec:handelsmaschine-selbsttest","text":""},{"seite":"handelsmaschine.html","titel":"21.12 Zusammenfassung","anker":"sec:handelsmaschine-zusammenfassung","text":""},{"seite":"synthese-anwendungen.html","titel":"Synthese Teil IV","anker":null,"text":""},{"seite":"synthese-anwendungen.html","titel":"Dieselbe Struktur, andere Namen","anker":"dieselbe-struktur-andere-namen","text":""},{"seite":"synthese-anwendungen.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-anwendungen.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-anwendungen.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"praxisfallen.html","titel":"Kapitel 22: Praxisfallen und der Weg zum produktiven Einsatz","anker":null,"text":""},{"seite":"praxisfallen.html","titel":"22.1 In 5 Minuten gelöst","anker":"sec:praxisfallen-schnellstart","text":"Die vorangegangenen vierzehn Kapitel enden fast alle mit derselben Empfehlung: Prüfen Sie das Ergebnis gegen die Wirklichkeit. Dieses Kapitel beginnt damit, diese Prüfung als wiede"},{"seite":"praxisfallen.html","titel":"22.2 Lernziele","anker":"sec:praxisfallen-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"praxisfallen.html","titel":"22.3 Die fünf typischen Praxisfallen","anker":"sec:praxisfallen-die-fuenf-typischen-praxisfallen","text":""},{"seite":"praxisfallen.html","titel":"Falle 1 — Infeasibility","anker":"falle-1-infeasibility","text":"Wenn Parameter unglücklich zusammentreffen — drei Mitarbeitende gleichzeitig krank, kein qualifizierter Ersatz —, meldet der Solver schlicht INFEASIBLE."},{"seite":"praxisfallen.html","titel":"Falle 2 — Der Black-Box-Effekt","anker":"falle-2-der-black-box-effekt","text":"Ein Planer oder Trader lehnt ein System ab, wenn er die Zuweisungen nicht nachvollziehen kann. Akzeptanz ist keine Nebensache, sondern Voraussetzung für den Betrieb."},{"seite":"praxisfallen.html","titel":"Falle 3 — Regimewechsel und Schätzfehler","anker":"falle-3-regimewechsel-und-schätzfehler","text":"Ein Modell, das auf einer mehrjährigen Aufwärtsphase kalibriert wurde, kennt keine Liquiditätskrise und keine Zinswende."},{"seite":"praxisfallen.html","titel":"Falle 4 — Lookahead- und Survivorship-Bias","anker":"falle-4-lookahead--und-survivorship-bias","text":"Ausführlich in Kapitel 21 behandelt. Die Kurzfassung: Beide Fehler machen Backtests systematisch zu gut, ohne eine Fehlermeldung zu erzeugen. Der Lookahead-Selbsttest aus Abschnitt"},{"seite":"praxisfallen.html","titel":"Falle 5 — Laufzeitexplosion","anker":"falle-5-laufzeitexplosion","text":"Exakte Optimalität kann bei MILP oder CP-SAT Minuten bis Stunden dauern — und die Laufzeit wächst nicht linear mit der Problemgröße."},{"seite":"praxisfallen.html","titel":"22.4 Constraint Attribution: welche Bedingung kostet wie viel?","anker":"sec:praxisfallen-attribution","text":"Der Constraint-Trace beantwortet „warum diese Zuweisung?“. Der Deletion Filter in Anhang C beantwortet „warum geht es gar nicht?“. Der häufigste Fall in der Praxis liegt dazwischen"},{"seite":"praxisfallen.html","titel":"Drei Befunde, und keiner steht im Solverergebnis","anker":"drei-befunde-und-keiner-steht-im-solverergebnis","text":"Erstens: bindend heißt nicht teuer. Fünf Bedingungen binden, aber „Lackierbudget Schicht 2“ tut es mit einem Schattenpreis von 0,00 €. Sie berührt den Plan, ohne ihn zu verteuern —"},{"seite":"praxisfallen.html","titel":"Vom Zahlenwerk zur Vorlage","anker":"vom-zahlenwerk-zur-vorlage","text":"Alle drei Antworten stehen bis hierhin in Tabellen. Tabellen liest niemand in einer Abteilungsleiterrunde. Teil (4) des Programms baut aus denselben Zahlen einen Text — und das ist"},{"seite":"praxisfallen.html","titel":"22.5 Architektur einer produktionsreifen OR-Plattform","anker":"sec:praxisfallen-architektur-einer-produktionsreifen-or-plattform","text":""},{"seite":"praxisfallen.html","titel":"22.6 Der gemeinsame Unterbau: or_kern.py","anker":"sec:praxisfallen-or-kern","text":"Die fünf Prinzipien oben beschreiben eine Plattform. Der Weg dorthin beginnt aber viel kleiner — mit der Frage, welche Teile eines Optimierungsprogramms immer dieselben sind."},{"seite":"praxisfallen.html","titel":"Die vier Schichten","anker":"die-vier-schichten","text":"\\underbrace{\\text{Rohdaten}}_{\\text{Excel, CSV, ERP}}\n\\;\\longrightarrow\\;\n\\underbrace{\\text{Domänenmodell}}_{\\text{geprüft, solverfrei}}\n\\;\\longrightarrow\\;\n\\underbrace{\\text{Model"},{"seite":"praxisfallen.html","titel":"Ein Enum für fünf Bibliotheken","anker":"ein-enum-für-fünf-bibliotheken","text":"Der unscheinbarste, aber wirksamste Teil ist die Statusübersetzung. Dieselbe Aussage heißt in den fünf im Buch verwendeten Bibliotheken:"},{"seite":"praxisfallen.html","titel":"22.7 Der Solverwechsel in der Praxis","anker":"sec:praxisfallen-solverwechsel","text":"Der vorige Abschnitt endet mit einer Behauptung: Ein Solverwechsel betreffe genau einen Baustein. Das ist die Art Satz, die in jedem Architekturvortrag vorkommt und die man deshalb"},{"seite":"praxisfallen.html","titel":"22.8 Eine Checkliste vor dem Produktivgang","anker":"sec:praxisfallen-eine-checkliste-vor-dem-produktivgang","text":"Arbeiten Sie diese Liste ab, bevor ein OR-System in Betrieb geht:"},{"seite":"praxisfallen.html","titel":"22.9 Weiterführende Literatur und Roadmap","anker":"sec:praxisfallen-weiterfuehrende-literatur-und-roadmap","text":"Grundlagen und lineare/ganzzahlige Optimierung * Bertsimas &amp; Tsitsiklis: Introduction to Linear Optimization — Standardwerk zu Simplex, Dualität, Polyedertheorie. * Wolsey: Int"},{"seite":"praxisfallen.html","titel":"22.10 Übungsaufgaben","anker":"sec:praxisfallen-uebungsaufgaben","text":""},{"seite":"praxisfallen.html","titel":"22.11 Finde den Denkfehler","anker":"sec:praxisfallen-denkfehler","text":"Falle 5 — die Laufzeitexplosion — zeigt sich im Betrieb selten als Absturz. Sie zeigt sich als gar nichts."},{"seite":"praxisfallen.html","titel":"22.12 Micro-Quiz","anker":"sec:praxisfallen-quiz","text":""},{"seite":"praxisfallen.html","titel":"22.13 Selbsttest","anker":"sec:praxisfallen-selbsttest","text":""},{"seite":"praxisfallen.html","titel":"22.14 Zusammenfassung","anker":"sec:praxisfallen-zusammenfassung","text":""},{"seite":"testing.html","titel":"Kapitel 23: Testen, Messen, Ausliefern","anker":null,"text":""},{"seite":"testing.html","titel":"23.1 In 5 Minuten gelöst","anker":"sec:testing-schnellstart","text":""},{"seite":"testing.html","titel":"23.2 Lernziele","anker":"sec:testing-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"testing.html","titel":"23.3 Warum Optimierungsmodelle schwer zu testen sind","anker":"sec:testing-warum-schwer","text":"Bei einer gewöhnlichen Funktion schreibt man den erwarteten Wert hin:"},{"seite":"testing.html","titel":"23.4 Die Testsuite","anker":"sec:testing-suite","text":""},{"seite":"testing.html","titel":"23.5 Wer testet die Tests?","anker":"sec:testing-mutation","text":"33 grüne Tests. Das ist noch kein Ergebnis."},{"seite":"testing.html","titel":"Der erste Lauf fand zwei Lücken","anker":"der-erste-lauf-fand-zwei-lücken","text":"Die Suite hatte zunächst 31 Tests. Der Mutationstest sagte dazu:"},{"seite":"testing.html","titel":"23.6 Ein Vergleich, dem man glauben kann","anker":"sec:testing-benchmark","text":"Solververgleiche stehen in jedem Blog, und die meisten sind wertlos. Nicht, weil falsch gemessen wurde, sondern weil zu wenig dazugesagt wird."},{"seite":"testing.html","titel":"Die interessanteste Spalte heißt „Anteil”","anker":"die-interessanteste-spalte-heißt-anteil","text":"Sie sagt, wie viel der Zeit in den Aufbau des Modells geht statt ins Lösen — und die Zahlen sind unbequem:"},{"seite":"testing.html","titel":"23.7 Das Modell als Dienst","anker":"sec:testing-dienst","text":"Ein Modell hinter eine HTTP-Schnittstelle zu hängen sieht nach einer Fingerübung aus. Es gibt aber einen Unterschied, der die ganze Bauform bestimmt:"},{"seite":"testing.html","titel":"Threads oder Prozesse? Eine Messung, keine Meinung","anker":"threads-oder-prozesse-eine-messung-keine-meinung","text":"Punkt 6 des Selbsttests misst, ob vier Solverläufe in Threads sich überlappen. Das Ergebnis ist ein Faktor von rund 3,5 bei vier Threads: OR-Tools rechnet in C++ und gibt den GIL w"},{"seite":"testing.html","titel":"Was im echten Betrieb dazukommt","anker":"was-im-echten-betrieb-dazukommt","text":"Das Beispiel hält das Auftragsbuch in einem dict und rechnet in Threads desselben Prozesses. Für einen einzelnen Rechner reicht das; darüber hinaus nicht:"},{"seite":"testing.html","titel":"23.8 Übungsaufgaben","anker":"sec:testing-uebungsaufgaben","text":""},{"seite":"testing.html","titel":"23.9 Finde den Denkfehler","anker":"sec:testing-denkfehler","text":""},{"seite":"testing.html","titel":"23.10 Micro-Quiz","anker":"sec:testing-quiz","text":""},{"seite":"testing.html","titel":"23.11 Selbsttest","anker":"sec:testing-selbsttest","text":""},{"seite":"testing.html","titel":"23.12 Zusammenfassung","anker":"sec:testing-zusammenfassung","text":""},{"seite":"synthese-praxis.html","titel":"Synthese Teil V","anker":null,"text":""},{"seite":"synthese-praxis.html","titel":"Was schiefgeht, und was dagegen hilft","anker":"was-schiefgeht-und-was-dagegen-hilft","text":""},{"seite":"synthese-praxis.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-praxis.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-praxis.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"projektwerkstatt.html","titel":"Projektwerkstatt","anker":null,"text":""},{"seite":"projektwerkstatt.html","titel":"Wie Sie ein Projekt bearbeiten","anker":"wie-sie-ein-projekt-bearbeiten","text":"Halten Sie sich an diese Reihenfolge — sie ist aus vielen gescheiterten und einigen gelungenen Projekten destilliert:"},{"seite":"projektwerkstatt.html","titel":"P1 — Vertretungsplaner für eine Schule","anker":"p1-vertretungsplaner-für-eine-schule","text":""},{"seite":"projektwerkstatt.html","titel":"P2 — Schichtplanung für ein Pflegeteam","anker":"p2-schichtplanung-für-ein-pflegeteam","text":""},{"seite":"projektwerkstatt.html","titel":"P3 — Tourenplanung für einen Lieferdienst","anker":"p3-tourenplanung-für-einen-lieferdienst","text":""},{"seite":"projektwerkstatt.html","titel":"P4 — Standort- und Lagernetzplanung","anker":"p4-standort--und-lagernetzplanung","text":""},{"seite":"projektwerkstatt.html","titel":"P5 — Produktionsplanung mit Rüstkosten","anker":"p5-produktionsplanung-mit-rüstkosten","text":""},{"seite":"projektwerkstatt.html","titel":"P6 — Portfolio-Rebalancer für ein Privatdepot","anker":"p6-portfolio-rebalancer-für-ein-privatdepot","text":""},{"seite":"projektwerkstatt.html","titel":"P7 — Risikoreport mit CVaR und Stresstests","anker":"p7-risikoreport-mit-cvar-und-stresstests","text":""},{"seite":"projektwerkstatt.html","titel":"P9 — Wenn der Solver aussteigt: Tourenplanung in Echtgröße","anker":"p9-wenn-der-solver-aussteigt-tourenplanung-in-echtgröße","text":""},{"seite":"projektwerkstatt.html","titel":"P10 — Zwei Ziele, eine Entscheidung: Kosten gegen CO₂","anker":"p10-zwei-ziele-eine-entscheidung-kosten-gegen-co₂","text":""},{"seite":"projektwerkstatt.html","titel":"P11 — Vom Skript zum Dienst: das Modell übergeben","anker":"p11-vom-skript-zum-dienst-das-modell-übergeben","text":""},{"seite":"projektwerkstatt.html","titel":"P8 — Freies Projekt aus Ihrem eigenen Umfeld","anker":"p8-freies-projekt-aus-ihrem-eigenen-umfeld","text":""},{"seite":"projektwerkstatt.html","titel":"Eine Bitte zum Schluss","anker":"eine-bitte-zum-schluss","text":"Wenn Sie ein Projekt fertig haben, machen Sie zwei Dinge:"},{"seite":"anhang-loesungen.html","titel":"Anhang A: Lösungen zu allen Übungsaufgaben","anker":null,"text":""},{"seite":"anhang-loesungen.html","titel":"A.1 Lösungen zu Kapitel „Einführung in Operations Research — Vom Ursprung zur mathematischen Entscheidungsfindung“","anker":"sec:loesungen-einfuehrung","text":"1.1 — Analytik-Stufen. (a) deskriptiv (Vergangenheit beschreiben). (b) prädiktiv (Prognose). (c) präskriptiv (Handlungsanweisung unter Restriktionen). (d) prädiktiv (Klassifikation"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Schreinerei verdoppelt ihren Gewinn","anker":"finde-den-denkfehler-die-schreinerei-verdoppelt-ihren-gewinn","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-einfuehrung","text":"1 — (c) präskriptiv. Die Frage lautet „was tun“, und es gibt eine Restriktion (das Restbudget). Dass man für die Bewertung eine Prognose braucht, macht die Frage nicht prädiktiv — "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-einfuehrung","text":""},{"seite":"anhang-loesungen.html","titel":"A.2 Lösungen zu Kapitel „Das mathematische Fundament — Vektoren, Matrizen, Konvexität“","anker":"sec:loesungen-fundament","text":"2.1 — Matrixform lesen. \\max 4x_1 + x_2 + 6x_3 u. d. N. x_1 + 2x_2 \\le 10, x_2 + 3x_3 \\le 12, x \\ge 0. 3 Variablen, 2 Nebenbedingungen (plus Nichtnegativität)."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die unauffällige Transposition","anker":"finde-den-denkfehler-die-unauffällige-transposition","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-fundament","text":"1 — (b). Der Fundamentalsatz sagt nur, dass ein Optimum in einer Ecke angenommen wird. (a) ist falsch, weil es mehrere optimale Ecken geben kann (wenn die Zielfunktion parallel zu "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-fundament","text":""},{"seite":"anhang-loesungen.html","titel":"A.3 Lösungen zu Kapitel „Das Python-Ökosystem für OR — Solver, Bindings und Modellierungsschichten“","anker":"sec:loesungen-oekosystem","text":"3.1 — Solverwahl. (a) CP-SAT (diskrete Zuordnung mit Zeitfenstern). (b) scipy.optimize.linprog (klassisches Mischungs-LP, klein). (c) CVXPY (konvexes QP). (d) MIQP — CVXPY mit Binä"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Der Solver, der angeblich dreimal schneller ist","anker":"finde-den-denkfehler-der-solver-der-angeblich-dreimal-schneller-ist","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-oekosystem","text":"1 — (b) CP-SAT. Es geht um diskrete Zuweisung mit logischen Regeln („nicht gleichzeitig“) — CP-SATs Kerngebiet, und Regeln dieser Art formuliert man dort direkt statt über Big-M-Um"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-oekosystem","text":""},{"seite":"anhang-loesungen.html","titel":"A.4 Lösungen zu Kapitel „Vom Management-Wunsch zum Modell“","anker":"sec:loesungen-modellierung","text":"4.1 — Die fehlende Lesart. Eine dritte Lesart: „Der Umsatzanteil der Stammkunden darf nicht unter 30 % fallen.” Sie zielt weder auf einzelne Aufträge noch auf einzelne Kunden, sond"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Die Engpassmaschine soll zu mindestens 92 % ausgelastet sein“","anker":"denkfehler-loesung-modellierung","text":"Fehler 1: Auslastung ist ein Ergebnis, keine Anforderung."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-modellierung","text":"1. b) Die Zielfunktion ist eine vollständige Aussage darüber, was zählt. Was nicht darin steht, ist für das Modell nicht „weniger wichtig”, sondern wertlos. (a) und (c) sind frei e"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-modellierung","text":""},{"seite":"anhang-loesungen.html","titel":"A.5 Lösungen zu Kapitel „Lineare Programmierung — Simplex, Dualität und Schattenpreise“","anker":"sec:loesungen-lp","text":"5.1 — Schlupf deuten. (a) Ressourcen 1 und 3 (Schlupf 0). (b) Ja: Überall ist s_i \\cdot y_i = 0. (c) In Ressource 3 — der höchste Schattenpreis (9,8) bedeutet den größten Grenznutz"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die 380 000-Euro-Maschine","anker":"finde-den-denkfehler-die-380-000-euro-maschine","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-lp","text":"1 — (b). Genau der Satz vom komplementären Schlupf: s_i \\cdot y_i = 0. Ist Schlupf vorhanden, muss der Dualwert null sein. (a) hat die Aussage in ihr Gegenteil verkehrt; (c) verwec"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-lp","text":""},{"seite":"anhang-loesungen.html","titel":"A.6 Lösungen zu Kapitel „Gemischt-ganzzahlige Optimierung — Diskrete Entscheidungen und Branch-and-Bound“","anker":"sec:loesungen-milp","text":"6.1 — Runden widerlegen. Beispiel: \\max x_1 + x_2 u. d. N. 10x_1 + 10x_2 \\le 15, ganzzahlig. LP-Optimum Z = 1{,}5; Abrunden ergibt (0,0) mit Z = 0 — 100 % Verlust. Das ganzzahlige "},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Elf Lager, die keine Fixkosten kosten","anker":"finde-den-denkfehler-elf-lager-die-keine-fixkosten-kosten","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-milp","text":"1 — (b). Der Gap (48\\,200 - 47\\,100)/48\\,200 = 2{,}3\\,\\% ist eine Garantie, keine Schätzung: Besser als 47 100 € kann es nachweislich nicht werden, also ist der vorliegende Plan hö"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-milp","text":""},{"seite":"anhang-loesungen.html","titel":"A.7 Lösungen zu Kapitel „Constraint Programming mit CP-SAT — Logik, Scheduling und Zuweisung“","anker":"sec:loesungen-cpsat","text":"7.1 — Propagation. Aus x_1 + x_2 = 8 und x_1 &lt; x_2 folgt x_1 &lt; 4, also x_1 \\in \\{2,3\\} (denn x_2 = 8-x_1 \\le 6 verlangt x_1 \\ge 2) und entsprechend x_2 \\in \\{5,6\\}. Aus 36 Ko"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Betriebsvereinbarung, die niemanden interessiert","anker":"finde-den-denkfehler-die-betriebsvereinbarung-die-niemanden-interessiert","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-cpsat","text":"1 — (b) wirkungslos. INFEASIBLE ist kein Abbruch, sondern ein Beweis: CP-SAT hat gezeigt, dass keine zulässige Lösung existiert. Mehr Zeit (a) oder mehr Arbeiter (c) ändern daran n"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-cpsat","text":""},{"seite":"anhang-loesungen.html","titel":"A.8 Lösungen zu Kapitel „Graphen, Flüsse und Touren — Min-Cost-Flow, Matching und VRP“","anker":"sec:loesungen-graphen","text":"8.1 — Flusserhaltung. b_i = (15+3) - (12+8) = -2 → Senke (Nettobedarf 2)."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die vergessene Dimension","anker":"finde-den-denkfehler-die-vergessene-dimension","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-graphen","text":"1 — (b) totale Unimodularität. Jede quadratische Teilmatrix hat Determinante 0, +1 oder -1; bei ganzzahliger rechter Seite sind deshalb alle Ecken des zulässigen Bereichs ganzzahli"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-graphen","text":""},{"seite":"anhang-loesungen.html","titel":"A.9 Lösungen zu Kapitel „Metaheuristiken — wenn der exakte Solver aussteigt“","anker":"sec:loesungen-metaheuristiken","text":"9.1 — Die Kurzsichtigkeit von Hand. Ab Schwarz wählt die Faustregel: Schwarz → Dunkelrot (26) → Rot (5) → … und muss dann in die helle Gruppe: Rot → Weiß (50) → Elfenbein (2) → Bei"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Unsere Heuristik ist 6 % besser“","anker":"denkfehler-loesung-metaheuristiken","text":"Es fehlt die Schranke."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-metaheuristiken","text":"1. b) Eine lokale Suche lebt von der Zahl geprüfter Züge. Die volle Summe kostet O(n) statt O(1) — bei 500 Aufträgen zwei Größenordnungen weniger Züge im selben Zeitbudget. Numeris"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-metaheuristiken","text":""},{"seite":"anhang-loesungen.html","titel":"A.10 Lösungen zu Kapitel „Spaltengenerierung“","anker":"sec:loesungen-dekomposition","text":"10.1 — Das Abbruchkriterium. Die Eins ist der Zielfunktionskoeffizient eines Musters: Jedes geschnittene Muster verbraucht genau eine Mutterrolle, und die Zielfunktion lautet \\min "},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Die LP-Lösung sagt 72,92 — also runden wir auf“","anker":"denkfehler-loesung-dekomposition","text":"Das Aufrunden kostet sechs Rollen — acht Prozent."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-dekomposition","text":"1. b) Das Mustermodell kennt keine einzelnen Rollen und damit keine austauschbaren Objekte — die Symmetrie verschwindet. (a) trifft zufällig auch zu, ist aber nicht der Grund; (c) "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-dekomposition","text":""},{"seite":"anhang-loesungen.html","titel":"A.11 Lösungen zu Kapitel „Quadratische und nichtlineare Optimierung — KKT, Lagrange, Konvexität“","anker":"sec:loesungen-qp-nlp","text":"11.1 — Konvexität einordnen. PSD (alle \\ge 0) → konvex, aber nicht streng konvex; die Lösung ist nicht notwendig eindeutig. Der Eigenwert 0 bedeutet eine flache Richtung: Entlang d"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Kovarianzmatrix aus dem Controlling","anker":"finde-den-denkfehler-die-kovarianzmatrix-aus-dem-controlling","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-qp-nlp","text":"1 — (b). Der DCPError ist eine inhaltliche Aussage: Für diese Formulierung kann es keine Optimalitätsgarantie geben. Auf scipy auszuweichen (a) beseitigt die Meldung, nicht die Urs"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-qp-nlp","text":""},{"seite":"anhang-loesungen.html","titel":"A.12 Lösungen zu Kapitel „Optimierung unter Unsicherheit — Monte-Carlo, Stochastik, Robustheit“","anker":"sec:loesungen-unsicherheit","text":"12.1 — Fluch des Durchschnitts. Weil sich Verzögerungen fortpflanzen und die Verteilung rechtsschief ist: Ein Gewerk, das früher fertig wird, beschleunigt selten den Gesamtablauf ("},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Warum jedes Projekt zu spät fertig wird","anker":"finde-den-denkfehler-warum-jedes-projekt-zu-spät-fertig-wird","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-unsicherheit","text":"1 — (b). Das kritische Verhältnis 1400/(1400+120) = 0{,}921 gibt an, wie weit man sich auf die günstigere Fehlerseite stellen soll: Bestellt wird das 92,1-%-Quantil des Bedarfs, hi"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-unsicherheit","text":""},{"seite":"anhang-loesungen.html","titel":"A.13 Lösungen zu Kapitel „Dynamische Programmierung — Die Bellman-Gleichung und Order-Execution“","anker":"sec:loesungen-dynamische-programmierung","text":"13.1 — Bausteine. Stufe = Tag 1–5; Zustand = verbleibende Distanz (ggf. plus Erschöpfungsgrad); Aktion = heutige Tagesetappe; Wertfunktion = minimale Restanstrengung. Nicht in den "},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Der Zustand, der zu wenig weiß","anker":"finde-den-denkfehler-der-zustand-der-zu-wenig-weiß","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-dynamische-programmierung","text":"1 — (b). V_t setzt V_{t+1} voraus: Eine Entscheidung lässt sich erst bewerten, wenn feststeht, was sie für die Zukunft bedeutet. Am Ende ist dieser Wert bekannt (V_T = g), deshalb "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-dynamische-programmierung","text":""},{"seite":"anhang-loesungen.html","titel":"A.14 Lösungen zu Kapitel „Mehrere Ziele — Pareto-Fronten statt Gewichte“","anker":"sec:loesungen-mehrziel","text":"14.1 — Dominanz prüfen. P (14 000 €, 6 000 kg) und Q (13 800 €, 6 100 kg): keiner dominiert den anderen. Q ist billiger, P ist sauberer — beide sind pareto-optimal, die Wahl ist ei"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Wir haben die Gewichte sauber kalibriert“","anker":"denkfehler-loesung-mehrziel","text":"Der Lösungsraum wurde nicht vollständig abgetastet — und zwar aus prinzipiellen Gründen."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-mehrziel","text":"1. b) Geometrisch heißt „gewichtete Summe minimieren“: eine Gerade von links unten an die Punktwolke schieben. Sie berührt immer einen Eckpunkt der unteren konvexen Hülle; Punkte d"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-mehrziel","text":""},{"seite":"anhang-loesungen.html","titel":"A.15 Lösungen zu Kapitel „Predict-then-Optimize“","anker":"sec:loesungen-prognose","text":"15.1 — Andere Preise. Der Überhang kostet nur noch 1,50 €, die Fehlmenge weiterhin 6 €. Das kritische Verhältnis steigt von 6/9 = 0{,}667 auf 6/7{,}5 = 0{,}80. Die Bestellmenge ver"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Wir haben die Prognose um 18 % verbessert“","anker":"denkfehler-loesung-prognose","text":"Der Sicherheitszuschlag stammt aus den Residuen des alten Modells."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-prognose","text":"1. b) Die Kostenasymmetrie verschiebt das Optimum vom Erwartungswert zum kritischen Quantil. (a) und (c) sind reale Probleme, aber nicht dieses: Selbst bei perfekt bekannter, norma"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-prognose","text":""},{"seite":"anhang-loesungen.html","titel":"A.16 Lösungen zu Kapitel „Die Strukturbrücke — dieselbe Mathematik, zwei Welten“","anker":"sec:loesungen-bruecke","text":"16.1 — Die dritte Ressource. „Liquidität” — wie schnell sich eine Position ohne Kursabschlag verkaufen lässt — entspricht in der Werkstatt der Umrüstbarkeit oder der Vorlaufzeit: W"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Das ist doch dasselbe Problem“","anker":"denkfehler-loesung-bruecke","text":"Fehler 1: Varianz ist hier das falsche Risikomaß — genau der Fall, für den es CVaR gibt."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-bruecke","text":"1. b) Beide Probleme haben dieselbe Struktur: knappe Größen auf konkurrierende Verwendungen verteilen. (a) ist falsch — Produktionsproblem ist streng typisiert und validiert beim E"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-bruecke","text":""},{"seite":"anhang-loesungen.html","titel":"A.17 Lösungen zu Kapitel „Supply-Chain und Energieeinsatz unter Unsicherheit“","anker":"sec:loesungen-supplychain","text":"17.1 — Der Umschlagpunkt. Bei 60 MW statt 120 MW halbiert sich die Grenzkostendifferenz je Stunde (83 €/MWh × 60 MW = 4 980 €/h). Der Umschlagpunkt verdoppelt sich auf 7,7 Stunden."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Wir rechnen mit dem P50-Szenario“","anker":"denkfehler-loesung-supplychain","text":"Fehler 1: Der Median ist keine Absicherung, sondern eine Wette auf die Hälfte der Fälle."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-supplychain","text":"1. b) Anfahrkosten und Mindestlaufzeiten koppeln die Stunden. Die Merit-Order ist eine Sortierung und kennt keine Kopplung über die Zeit; sie ist für einen einzelnen Zeitpunkt rich"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-supplychain","text":""},{"seite":"anhang-loesungen.html","titel":"A.18 Lösungen zu Kapitel „Finanzdaten-Modellierung — Renditen, Kovarianz und Shrinkage“","anker":"sec:loesungen-finanzdaten","text":"18.1 — Renditeart. (a) diskret. (b) logarithmisch. (c) logarithmisch (bzw. diskret kumuliert über Produkt). (d) logarithmisch (statistische Eigenschaften)."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Das risikofreie Portfolio","anker":"finde-den-denkfehler-das-risikofreie-portfolio","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-finanzdaten","text":"1 — (b) bei 75 %. 1{,}0 \\cdot 0{,}5 \\cdot 1{,}5 = 0{,}75. Prozentuale Änderungen verknüpfen sich multiplikativ; ein Verlust wiegt schwerer als ein gleich großer Gewinn, weil er von"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-finanzdaten","text":""},{"seite":"anhang-loesungen.html","titel":"A.19 Lösungen zu Kapitel „Die moderne Portfoliotheorie nach Markowitz“","anker":"sec:loesungen-markowitz","text":"19.1 — Diversifikationseffekt."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Zwölf gleiche Anlagen, ein sehr ungleiches Portfolio","anker":"finde-den-denkfehler-zwölf-gleiche-anlagen-ein-sehr-ungleiches-portfolio","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-markowitz","text":"1 — (b). Das renditestärkste Portfolio auf der Effizienzlinie hat notwendig das schlechteste Rendite-Risiko-Verhältnis (im Kapitelbeispiel 0,44 gegenüber 0,75 beim Optimum) und bes"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-markowitz","text":""},{"seite":"anhang-loesungen.html","titel":"A.20 Lösungen zu Kapitel „Tail-Risiko, CVaR und Transaktionskosten“","anker":"sec:loesungen-cvar","text":"20.1 — VaR und CVaR von Hand. Sortierte Verluste: -2{,}0; -1{,}2; -0{,}8; -0{,}4; 0{,}3; 0{,}6; 1{,}1; 2{,}1; 5{,}4; 8{,}9. Das 80 %-Quantil ist der 8. Wert: \\text{VaR}_{80\\%} = 2{"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Das Risikobudget, das durch Diversifikation stieg","anker":"finde-den-denkfehler-das-risikobudget-das-durch-diversifikation-stieg","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-cvar","text":"1 — (b) nichts. Der VaR ist ein Quantil: Er markiert die Schwelle und sagt nichts über den Bereich dahinter. Genau das zeigt der Schnellstart dieses Kapitels — zwei Anlagen mit ide"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-cvar","text":""},{"seite":"anhang-loesungen.html","titel":"A.21 Lösungen zu Kapitel „Die vollständige quantitative Handelsmaschine“","anker":"sec:loesungen-handelsmaschine","text":"21.1 — Lookahead erkennen. (a) sauber (:heute). (b) Lookahead — gesamter Zeitraum. (c) Lookahead — Volatilität über den ganzen Zeitraum enthält Zukunft. (d) sauber — Gewichte aus D"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Strategie mit dem hochsignifikanten Ergebnis","anker":"finde-den-denkfehler-die-strategie-mit-dem-hochsignifikanten-ergebnis","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-handelsmaschine","text":"1 — (b) nichts. Beide Kennzahlen sind Mittelwerte über die Zeit und blind für die Reihenfolge. Der Schnellstart dieses Kapitels zeigt es an denselben Tagesrenditen, nur anders ange"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-handelsmaschine","text":""},{"seite":"anhang-loesungen.html","titel":"A.22 Lösungen zu Kapitel „Praxisfallen und der Weg zum produktiven Einsatz“","anker":"sec:loesungen-praxisfallen","text":"22.1 — Hart oder weich, revisited. (a) hart: gesetzliche Ruhezeit, Qualifikationspflicht. (b) weich: individuelle Wunschtage, Vermeidung von Freistunden. (c) diskutabel: Höchstzahl"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Das Modell, das seit einem Jahr nicht mehr optimiert","anker":"finde-den-denkfehler-das-modell-das-seit-einem-jahr-nicht-mehr-optimiert","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-praxisfallen","text":"1 — (b) das Zeitlimit greift. Eine Laufzeit, die exakt dem Limit entspricht und über Monate konstant bleibt, ist kein Stabilitätsbeleg, sondern zeigt, dass der Solver jedes Mal abg"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-praxisfallen","text":""},{"seite":"anhang-loesungen.html","titel":"A.23 Lösungen zu Kapitel „Testen, Messen, Ausliefern“","anker":"sec:loesungen-testing","text":"23.1 — Den Schnellstart reparieren. Zwei Wege: (a) Abrunden statt runden — np.floor statt np.round. Das ist immer zulässig, weil weniger produzieren nie eine Kapazität sprengt, kos"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Die Suite ist grün, das Modell stimmt”","anker":"denkfehler-loesung-testing","text":"Alle Prüfungen benutzen dieselbe falsche Zahl."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-testing","text":"1. b) Es gibt keine unabhängige Quelle für die richtige Antwort — sonst bräuchte man den Solver nicht. (a) ist bei festgelegtem Seed und Zeitlimit meist beherrschbar; (c) löst man "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-testing","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"Anhang B: Katalog der Modellierungsmuster","anker":null,"text":""},{"seite":"anhang-modellierungsmuster.html","titel":"Übersicht","anker":"übersicht","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"Logische Schalter","anker":"logische-schalter","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B1 — Aktivierungsschalter (Fixkosten)","anker":"b1-aktivierungsschalter-fixkosten","text":"Regel. „Wenn überhaupt etwas produziert wird, fallen Rüstkosten F an.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B2 — Semikontinuierliche Variable","anker":"b2-semikontinuierliche-variable","text":"Regel. „Entweder gar nicht oder mindestens L (und höchstens U).“"},{"seite":"anhang-modellierungsmuster.html","titel":"B3 — Implikation","anker":"b3-implikation","text":"Regel. „Wenn A gewählt wird, muss auch B gewählt werden.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B4 — Entweder-Oder (disjunktive Bedingung)","anker":"b4-entweder-oder-disjunktive-bedingung","text":"Regel. „Es muss f(\\mathbf{x}) \\le b_1 oder g(\\mathbf{x}) \\le b_2 gelten.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B5 — Exklusiv-Oder","anker":"b5-exklusiv-oder","text":"Regel. „Genau eine der Optionen wird gewählt.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B6 — Kardinalität","anker":"b6-kardinalität","text":"Regel. „Höchstens (mindestens, genau) K von N.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B7 — Bedingte Kopplung (Konjunktion)","anker":"b7-bedingte-kopplung-konjunktion","text":"Regel. „Wenn A und B, dann auch C.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Mengen und Grenzen","anker":"mengen-und-grenzen","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B8 — Weiche Grenze mit Strafkosten","anker":"b8-weiche-grenze-mit-strafkosten","text":"Regel. „Möglichst nicht über b — wenn doch, kostet es.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B9 — Gestaffelte Preise (stückweise linear)","anker":"b9-gestaffelte-preise-stückweise-linear","text":"Regel. „Die ersten 100 Stück kosten 5 €, danach 8 €.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B10 — Absolutbetrag / Abweichung","anker":"b10-absolutbetrag-abweichung","text":"Regel. „Die Abweichung vom Zielwert soll klein sein, egal in welche Richtung.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B11 — Min/Max in der Zielfunktion","anker":"b11-minmax-in-der-zielfunktion","text":"Regel. „Der am schlechtesten gestellte Beteiligte soll möglichst gut dastehen“ (Maximin/Fairness)."},{"seite":"anhang-modellierungsmuster.html","titel":"B12 — Verhältnis-Bedingung","anker":"b12-verhältnis-bedingung","text":"Regel. „Der Anteil von Gruppe G soll mindestens 30 % betragen.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B25 — Mindestabnahmemenge über einen Zeitraum","anker":"b25-mindestabnahmemenge-über-einen-zeitraum","text":"Regel. „Entweder wir arbeiten mit diesem Lieferanten gar nicht, oder wir nehmen ihm im Jahr mindestens 500 Einheiten ab.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B27 — Budgetlimit","anker":"b27-budgetlimit","text":"Regel. „Alle Maßnahmen zusammen dürfen 2 Mio. € nicht überschreiten.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Zeit und Reihenfolge","anker":"zeit-und-reihenfolge","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B13 — Vorrangbeziehung","anker":"b13-vorrangbeziehung","text":"Regel. „Arbeitsgang B darf erst beginnen, wenn A fertig ist.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B14 — Nichtüberlappung","anker":"b14-nichtüberlappung","text":"Regel. „Eine Maschine bearbeitet nur einen Job gleichzeitig.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B15 — Kumulative Ressource","anker":"b15-kumulative-ressource","text":"Regel. „Zu keinem Zeitpunkt dürfen mehr als 3 Arbeiten gleichzeitig laufen“ bzw. „die Stromlast darf 500 kW nie übersteigen“."},{"seite":"anhang-modellierungsmuster.html","titel":"B16 — Gleitendes Fenster","anker":"b16-gleitendes-fenster","text":"Regel. „Höchstens 5 Arbeitstage in Folge“ / „mindestens 2 freie Tage je Woche“."},{"seite":"anhang-modellierungsmuster.html","titel":"B17 — Umrüst- bzw. Wechselkosten","anker":"b17-umrüst--bzw.-wechselkosten","text":"Regel. „Ein Produktwechsel kostet Rüstzeit.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B26 — Rüstzeit als Kapazitätsverbrauch","anker":"b26-rüstzeit-als-kapazitätsverbrauch","text":"Regel. „Das Umrüsten kostet nicht nur Geld, es kostet Maschinenstunden — und die fehlen dann für die Produktion.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Robustheit und Diagnose","anker":"robustheit-und-diagnose","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B18 — Schlupfvariablen gegen Unlösbarkeit","anker":"b18-schlupfvariablen-gegen-unlösbarkeit","text":"Das wichtigste Muster für den Produktivbetrieb."},{"seite":"anhang-modellierungsmuster.html","titel":"B19 — Hierarchische Ziele (lexikografisch)","anker":"b19-hierarchische-ziele-lexikografisch","text":"Regel. „Erst die Besetzung sicherstellen, dann die Fairness optimieren.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B20 — Symmetriebrechung","anker":"b20-symmetriebrechung","text":"Regel. „Drei identische Maschinen — der Solver soll nicht alle Vertauschungen durchprobieren.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B21 — Worst-Case-Abzug (robuste Formulierung)","anker":"b21-worst-case-abzug-robuste-formulierung","text":"Regel. „Rechne nicht mit dem geschätzten Wert, sondern mit dem ungünstigsten plausiblen.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Netzwerke","anker":"netzwerke","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B22 — Flusserhaltung","anker":"b22-flusserhaltung","text":"\\sum_{j:(i,j)\\in E} x_{ij} - \\sum_{k:(k,i)\\in E} x_{ki} = b_i \\qquad \\forall i"},{"seite":"anhang-modellierungsmuster.html","titel":"B23 — Zuordnung 1:1","anker":"b23-zuordnung-11","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B24 — Subtour-Eliminierung","anker":"b24-subtour-eliminierung","text":"MTZ (einfach, aber schwach): u_i - u_j + C x_{ij} \\le C - d_j \\qquad \\forall i\\ne j"},{"seite":"anhang-modellierungsmuster.html","titel":"Ein Wort zur Auswahl","anker":"ein-wort-zur-auswahl","text":"Wenn mehrere Muster passen, entscheiden Sie nach dieser Reihenfolge:"},{"seite":"anhang-fehlerdiagnose.html","titel":"Anhang C: Fehlerdiagnose-Handbuch","anker":null,"text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"Symptom-Schnellübersicht","anker":"symptom-schnellübersicht","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C1 — INFEASIBLE","anker":"c1-infeasible","text":"Bedeutung. Es gibt keinen einzigen Punkt, der alle Bedingungen gleichzeitig erfüllt. Das ist eine Aussage über Ihr Modell, nicht über den Solver."},{"seite":"anhang-fehlerdiagnose.html","titel":"Diagnose in fünf Schritten","anker":"diagnose-in-fünf-schritten","text":"Schritt 1 — Trivialprüfungen von Hand. Rechnen Sie die offensichtlichen Bilanzen nach, bevor Sie den Solver befragen:"},{"seite":"anhang-fehlerdiagnose.html","titel":"Wenn die fünf Schritte nicht reichen: den Konflikt einkreisen","anker":"wenn-die-fünf-schritte-nicht-reichen-den-konflikt-einkreisen","text":"Warum Schritt 2 so oft ins Leere läuft. Bedingungen einzeln abzuschalten funktioniert nur, solange es einen Schuldigen gibt. Überlagern sich zwei unabhängige Widersprüche, bleibt d"},{"seite":"anhang-fehlerdiagnose.html","titel":"C2 — UNBOUNDED","anker":"c2-unbounded","text":"Bedeutung. Die Zielfunktion lässt sich unbegrenzt verbessern."},{"seite":"anhang-fehlerdiagnose.html","titel":"C3 — Zu langsam","anker":"c3-zu-langsam","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"Diagnosereihenfolge","anker":"diagnosereihenfolge","text":"1. Ist es überhaupt das Lösen?"},{"seite":"anhang-fehlerdiagnose.html","titel":"C4 — Unsinniges Ergebnis","anker":"c4-unsinniges-ergebnis","text":"Der Solver hat immer recht — bezogen auf das Modell, das Sie ihm gegeben haben. Wenn das Ergebnis unsinnig ist, beschreibt Ihr Modell nicht das Problem, das Sie meinen."},{"seite":"anhang-fehlerdiagnose.html","titel":"Checkliste","anker":"checkliste","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"Die wirksamste Gegenmaßnahme","anker":"die-wirksamste-gegenmaßnahme","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C5 — Instabile Lösung","anker":"c5-instabile-lösung","text":"Symptom. Ein zusätzlicher Handelstag, und das Portfolio sieht völlig anders aus."},{"seite":"anhang-fehlerdiagnose.html","titel":"Derselbe Fehler außerhalb der Finanzwelt","anker":"derselbe-fehler-außerhalb-der-finanzwelt","text":"Instabilität ist kein Portfolioproblem, sondern ein Skalierungsproblem — und es trifft jedes Modell, in dem Größen sehr verschiedener Ordnung nebeneinanderstehen. Euro-Beträge (10^"},{"seite":"anhang-fehlerdiagnose.html","titel":"C6 — Falsche Dualwerte","anker":"c6-falsche-dualwerte","text":"Symptom. Schattenpreise sind negativ, wo sie positiv sein sollten — oder null bei einer offensichtlich knappen Ressource."},{"seite":"anhang-fehlerdiagnose.html","titel":"Die drei Prüfungen","anker":"die-drei-prüfungen","text":"1. Vorzeichenkonvention. Haben Sie zur Maximierung negiert?"},{"seite":"anhang-fehlerdiagnose.html","titel":"C7 — Widersprüchliche Solver","anker":"c7-widersprüchliche-solver","text":"Symptom. linprog sagt 530, CVXPY sagt 529,8."},{"seite":"anhang-fehlerdiagnose.html","titel":"C8 — DCPError","anker":"c8-dcperror","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C9 — Importfehler","anker":"c9-importfehler","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C10 — Verdächtig guter Backtest","anker":"c10-verdächtig-guter-backtest","text":"Faustregel: Eine Sharpe Ratio über 2 bei einer einfachen Strategie ist fast immer ein Fehler, kein Fund."},{"seite":"anhang-fehlerdiagnose.html","titel":"Prüfreihenfolge","anker":"prüfreihenfolge","text":"1. Lookahead-Selbsttest."},{"seite":"anhang-fehlerdiagnose.html","titel":"C11 — Vertauschte Spalten","anker":"c11-vertauschte-spalten","text":"Symptom. Ergebnisse sind plausibel, aber falsch beschriftet — oder eine Sektorgrenze greift auf die falschen Titel."},{"seite":"anhang-fehlerdiagnose.html","titel":"Die goldene Diagnoseregel","anker":"die-goldene-diagnoseregel","text":""},{"seite":"anhang-spickzettel.html","titel":"Anhang D: Spickzettel der Solver","anker":null,"text":""},{"seite":"anhang-spickzettel.html","titel":"D1 — SciPy: linprog und milp","anker":"d1-scipy-linprog-und-milp","text":"Wofür. Die Einstiegsschicht: keine zusätzliche Installation, HiGHS als Unterbau, ideal für lineare und gemischt-ganzzahlige Probleme in Matrixform. Wofür nicht: alles, was sich nic"},{"seite":"anhang-spickzettel.html","titel":"Lineares Programm","anker":"lineares-programm","text":""},{"seite":"anhang-spickzettel.html","titel":"Ganzzahlig: milp","anker":"ganzzahlig-milp","text":""},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine","text":""},{"seite":"anhang-spickzettel.html","titel":"D2 — HiGHS über highspy","anker":"d2-highs-über-highspy","text":"Wofür. Derselbe Solver wie unter SciPy, aber direkt gesteuert: Optionen, Warm-Starts, inkrementelles Ändern eines bestehenden Modells. Wofür nicht: schnelles Hinschreiben — die CSR"},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-1","text":""},{"seite":"anhang-spickzettel.html","titel":"D3 — OR-Tools: pywraplp","anker":"d3-or-tools-pywraplp","text":"Wofür. Bequeme algebraische Schreibweise für LP und MILP mit umschaltbarem Backend (GLOP, SCIP, CBC, SAT). Wofür nicht: Scheduling und kombinatorische Bedingungen — dafür ist CP-SA"},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-2","text":""},{"seite":"anhang-spickzettel.html","titel":"D4 — CP-SAT: cp_model","anker":"d4-cp-sat-cp_model","text":"Wofür. Scheduling, Zuordnung, Reihenfolgen, alles Kombinatorische mit globalen Bedingungen. Wofür nicht: stetige Größen — CP-SAT rechnet ausschließlich ganzzahlig. Wer Nachkommaste"},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-3","text":""},{"seite":"anhang-spickzettel.html","titel":"D5 — CVXPY","anker":"d5-cvxpy","text":"Wofür. Konvexe Probleme: quadratische Ziele, Normen, CVaR, alles mit Regularisierungstermen. Wofür nicht: große kombinatorische Modelle — der Aufbau der Ausdrücke wird dann selbst "},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-4","text":""},{"seite":"anhang-spickzettel.html","titel":"D6 — Modellierungssprachen: Pyomo und Linopy","anker":"d6-modellierungssprachen-pyomo-und-linopy","text":"Wofür. Beide trennen Modell von Solver: dasselbe Modell läuft ohne Änderung unter HiGHS, CBC, Gurobi. Pyomo denkt in Mengen und Indizes wie eine mathematische Formulierung; Linopy "},{"seite":"anhang-spickzettel.html","titel":"Pyomo","anker":"pyomo","text":""},{"seite":"anhang-spickzettel.html","titel":"Linopy","anker":"linopy","text":""},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-5","text":""},{"seite":"anhang-spickzettel.html","titel":"Die gemeinsame Regel","anker":"die-gemeinsame-regel","text":"Alle sechs Seiten haben denselben längsten Abschnitt: die Statusauswertung. Das ist kein Zufall. Ein Solveraufruf hat nie zwei Ausgänge, sondern mindestens fünf — optimal, zulässig"},{"seite":"anhang-glossar.html","titel":"Anhang E: Glossar","anker":null,"text":""},{"seite":"anhang-literatur.html","titel":"Anhang F: Literaturverzeichnis","anker":null,"text":""},{"seite":"anhang-literatur.html","titel":"Grundlagen und Standardwerke des Operations Research","anker":"grundlagen-und-standardwerke-des-operations-research","text":"Zu Kapitel 1 und Kapitel 2."},{"seite":"anhang-literatur.html","titel":"Modellierung in Python: Bibliotheken und Modellierungsschichten","anker":"modellierung-in-python-bibliotheken-und-modellierungsschichten","text":"Zu Kapitel 3."},{"seite":"anhang-literatur.html","titel":"Lineare und ganzzahlige Optimierung","anker":"lineare-und-ganzzahlige-optimierung","text":"Zu Kapitel 5 und Kapitel 6."},{"seite":"anhang-literatur.html","titel":"Moderne Solver: Implementierung, Vergleich und Benchmarks","anker":"moderne-solver-implementierung-vergleich-und-benchmarks","text":"Zu Kapitel 3 und Anhang D. Neuer Abschnitt: Das Buch vergleicht durchgehend Solver, die Literatur dazu stand bisher verstreut."},{"seite":"anhang-literatur.html","titel":"Constraint Programming und Scheduling","anker":"constraint-programming-und-scheduling","text":"Zu Kapitel 7."},{"seite":"anhang-literatur.html","titel":"Graphen, Flüsse und Tourenplanung","anker":"graphen-flüsse-und-tourenplanung","text":"Zu Kapitel 8. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Metaheuristiken","anker":"metaheuristiken","text":"Zu Kapitel 9. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Spaltengenerierung und Dekomposition","anker":"spaltengenerierung-und-dekomposition","text":"Zu Kapitel 10. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Konvexe und nichtlineare Optimierung","anker":"konvexe-und-nichtlineare-optimierung","text":"Zu Kapitel 11."},{"seite":"anhang-literatur.html","titel":"Optimierung unter Unsicherheit: stochastisch, robust, Chance Constraints","anker":"optimierung-unter-unsicherheit-stochastisch-robust-chance-constraints","text":"Zu Kapitel 12 und Kapitel 17. Aus dem früheren Block „Konvexe, nichtlineare und robuste Optimierung” herausgelöst und erweitert."},{"seite":"anhang-literatur.html","titel":"Dynamische Programmierung und sequenzielle Entscheidungen","anker":"dynamische-programmierung-und-sequenzielle-entscheidungen","text":"Zu Kapitel 13. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Mehrkriterielle Optimierung","anker":"mehrkriterielle-optimierung","text":"Zu Kapitel 14. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Lernbasierte Optimierung: maschinelles Lernen und OR","anker":"lernbasierte-optimierung-maschinelles-lernen-und-or","text":"Zu Kapitel 15. Neuer Abschnitt — im Buch als eigenes Kapitel behandelt, in der Literaturliste bisher gar nicht vertreten."},{"seite":"anhang-literatur.html","titel":"Supply Chain, Energie und Produktionsplanung","anker":"supply-chain-energie-und-produktionsplanung","text":"Zu Kapitel 16 und Kapitel 17. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Quantitative Finanzmathematik","anker":"quantitative-finanzmathematik","text":"Zu Kapitel 18, Kapitel 19, Kapitel 20 und Kapitel 21."},{"seite":"anhang-literatur.html","titel":"Praxis: Betrieb, Test und Auslieferung von Optimierungsmodellen","anker":"praxis-betrieb-test-und-auslieferung-von-optimierungsmodellen","text":"Zu Kapitel 22 und Kapitel 23. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Software, Dokumentation und Datenquellen","anker":"software-dokumentation-und-datenquellen","text":""},{"seite":"anhang-literatur.html","titel":"Verbände, Normen und Regulierung","anker":"verbände-normen-und-regulierung","text":""},{"seite":"anhang-literatur.html","titel":"Änderungsprotokoll","anker":"änderungsprotokoll","text":"Stand: September 2026. Überarbeitung des Literaturverzeichnisses gegenüber der Vorgängerfassung."},{"seite":"anhang-literatur.html","titel":"Prüfung gegen Crossref","anker":"prüfung-gegen-crossref","text":"Alle Einträge wurden einzeln gegen die Crossref-API geprüft — bei einem Literaturverzeichnis ist eine erfundene oder falsch zitierte Quelle der schwerste Fehler, weil der Leser sie"}];
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Dabei kamen unter anderem textbasierte Auszeichnungsformate und Werkzeuge wie Markdown, Pandoc und LaTeX zum Einsatz; auch bei einzelnen technischen und redaktionellen Arbeitsschritten wurden KI-Systeme unterstützend verwendet.","anker":"die-erstellung-der-quelldateien-programmbeispiele-des-glossars-stichwortverzeichnisses-sowie-der-online--und-pdf-fassung-erfolgte-in-einer-linux-basierten-arbeitsumgebung.-dabei-kamen-unter-anderem-textbasierte-auszeichnungsformate-und-werkzeuge-wie-markdown-pandoc-und-latex-zum-einsatz-auch-bei-einzelnen-technischen-und-redaktionellen-arbeitsschritten-wurden-ki-systeme-unterstützend-verwendet.","text":""},{"seite":"notation.html","titel":"Notation & Abkürzungen","anker":null,"text":""},{"seite":"notation.html","titel":"1. Die Symbole der Optimierung","anker":"die-symbole-der-optimierung","text":""},{"seite":"notation.html","titel":"1.1 Grundgrößen jedes Modells","anker":"grundgrößen-jedes-modells","text":""},{"seite":"notation.html","titel":"1.2 Mengen und Zahlbereiche","anker":"mengen-und-zahlbereiche","text":""},{"seite":"notation.html","titel":"1.3 Operatoren und Schreibweisen","anker":"operatoren-und-schreibweisen","text":""},{"seite":"notation.html","titel":"1.4 Symbole der Finanzkapitel (Teil IV)","anker":"symbole-der-finanzkapitel-teil-iv","text":""},{"seite":"notation.html","titel":"1.5 Symbole der Verfahrenskapitel (Teil II und III)","anker":"symbole-der-verfahrenskapitel-teil-ii-und-iii","text":"Diese Zeichen kommen erst in den Kapiteln vor, die über das klassische LP hinausgehen."},{"seite":"notation.html","titel":"2. Abkürzungen — vollständig ausgeschrieben","anker":"abkürzungen-vollständig-ausgeschrieben","text":""},{"seite":"notation.html","titel":"2.1 Problemklassen und Verfahren","anker":"problemklassen-und-verfahren","text":""},{"seite":"notation.html","titel":"2.2 Probleme mit Eigennamen","anker":"probleme-mit-eigennamen","text":""},{"seite":"notation.html","titel":"2.3 Finanz- und Risikokennzahlen","anker":"finanz--und-risikokennzahlen","text":""},{"seite":"notation.html","titel":"2.4 Software, Solver und Schnittstellen","anker":"software-solver-und-schnittstellen","text":""},{"seite":"notation.html","titel":"2.5 Begriffe aus dem Solver-Alltag","anker":"begriffe-aus-dem-solver-alltag","text":""},{"seite":"notation.html","titel":"3. Konventionen in diesem Buch","anker":"konventionen-in-diesem-buch","text":"Vektoren sind grundsätzlich Spaltenvektoren und werden fett gesetzt: \\mathbf{x}. Ein transponierter Vektor \\mathbf{x}^\\top ist eine Zeile. Skalare bleiben mager: n, Z."},{"seite":"einfuehrung.html","titel":"Kapitel 1: Einführung in Operations Research — Vom Ursprung zur mathematischen Entscheidungsfindung","anker":null,"text":""},{"seite":"einfuehrung.html","titel":"1.1 In 5 Minuten gelöst","anker":"sec:einfuehrung-schnellstart","text":"Bevor wir über Theorie sprechen, lösen Sie ein echtes Problem. Kopieren Sie den folgenden Block, führen Sie ihn aus, sehen Sie das Ergebnis — die Erklärung kommt danach."},{"seite":"einfuehrung.html","titel":"1.2 Lernziele","anker":"sec:einfuehrung-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"einfuehrung.html","titel":"1.3 Was ist Operations Research wirklich?","anker":"sec:einfuehrung-was-ist-operations-research-wirklich","text":"Im Alltag lösen wir Probleme intuitiv: durch Ausprobieren, Daumenregeln, Erfahrung. Das funktioniert erstaunlich gut — bis eine bestimmte Schwelle überschritten wird. Dann bricht d"},{"seite":"einfuehrung.html","titel":"Die drei Stufen der Analytik","anker":"die-drei-stufen-der-analytik","text":"OR beantwortet nicht die beschreibende Frage „Was ist passiert?“ und nicht die vorhersagende Frage „Was wird passieren?“, sondern die vorschreibende (präskriptive) Kernfrage:"},{"seite":"einfuehrung.html","titel":"1.4 Warum Ausprobieren scheitert — mit eigener Rechnung","anker":"sec:einfuehrung-warum-ausprobieren-scheitert-mit-eigener-rechnung","text":"Die Behauptung „vollständiges Durchprobieren ist unmöglich“ wird oft aufgestellt und selten belegt. Rechnen wir sie einmal selbst nach — das Ergebnis prägt sich besser ein als jede"},{"seite":"einfuehrung.html","titel":"1.5 Historischer Kontext und Evolution","anker":"sec:einfuehrung-historischer-kontext-und-evolution","text":"Der Begriff entstand im Vorfeld des Zweiten Weltkriegs in Großbritannien. Militärische Führungsstäbe standen vor neuartigen logistischen und strategischen Fragen: Wie platziert man"},{"seite":"einfuehrung.html","titel":"1.6 Die vier universellen Bausteine jedes OR-Problems","anker":"sec:einfuehrung-die-vier-universellen-bausteine-jedes-or","text":"Dies ist der wichtigste Abschnitt des Kapitels. Jedes Optimierungsproblem der Welt lässt sich — unabhängig von der Branche — auf vier Elemente reduzieren. Wenn Sie später vor einem"},{"seite":"einfuehrung.html","titel":"Baustein 1 — Entscheidungsvariablen (x)","anker":"baustein-1-entscheidungsvariablen-x","text":"Die Stellschrauben des Modells: Werte, die der Algorithmus aktiv festlegen darf."},{"seite":"einfuehrung.html","titel":"Baustein 2 — Parameter / Eingabedaten (c, A, b)","anker":"baustein-2-parameter-eingabedaten-c-a-b","text":"Die unveränderlichen Fakten der Realität: historische Renditen, Maschinenstundensätze, Lagerkapazitäten, Steuerquoten, Entfernungen, Qualifikationen."},{"seite":"einfuehrung.html","titel":"Baustein 3 — Zielfunktion (f(x))","anker":"baustein-3-zielfunktion-fx","text":"Ein mathematischer Ausdruck, der die Güte einer Lösung zu einer einzigen Zahl verdichtet:"},{"seite":"einfuehrung.html","titel":"Baustein 4 — Nebenbedingungen (Constraints)","anker":"baustein-4-nebenbedingungen-constraints","text":"Regeln, die den Raum der erlaubten Lösungen einschränken:"},{"seite":"einfuehrung.html","titel":"Die vier Bausteine als eine Formel","anker":"die-vier-bausteine-als-eine-formel","text":"Setzt man die vier Bausteine zusammen, entsteht die Standardform, in der Ihnen jedes Optimierungsmodell dieses Buches begegnen wird. Sie sieht auf den ersten Blick sperrig aus — de"},{"seite":"einfuehrung.html","titel":"Die Bausteine in der Praxis: eine Vorlage","anker":"die-bausteine-in-der-praxis-eine-vorlage","text":"Damit die Zerlegung zur Gewohnheit wird, hier eine Vorlage, die Sie bei jedem neuen Problem ausfüllen können — vor der ersten Codezeile."},{"seite":"einfuehrung.html","titel":"1.7 Erstes Python-Vollbeispiel: Ressourcenallokation im Rechenzentrum","anker":"sec:einfuehrung-erstes-python-vollbeispiel-ressourcenallokation-im-rechenzentrum","text":""},{"seite":"einfuehrung.html","titel":"Szenario","anker":"szenario","text":"Ein quantitatives Handelshaus betreibt Serverkapazitäten für zwei Algorithmentypen:"},{"seite":"einfuehrung.html","titel":"Mathematische Formulierung","anker":"mathematische-formulierung","text":"Entscheidungsvariablen:"},{"seite":"einfuehrung.html","titel":"Umsetzung mit Google OR-Tools","anker":"umsetzung-mit-google-or-tools","text":""},{"seite":"einfuehrung.html","titel":"1.8 Von Excel zu Python: Ihre Daten liegen schon da","anker":"sec:einfuehrung-excel-bruecke","text":"In den allermeisten Betrieben liegen die Zahlen, die ein Optimierungsmodell braucht, nicht in einer Datenbank, sondern in einer Tabellenkalkulation. Und viele Leserinnen und Leser "},{"seite":"einfuehrung.html","titel":"1.9 Übungsaufgaben","anker":"sec:einfuehrung-uebungsaufgaben","text":""},{"seite":"einfuehrung.html","titel":"1.10 Finde den Denkfehler","anker":"sec:einfuehrung-denkfehler","text":"Ein Programm, das abstürzt, ist harmlos — Sie sehen sofort, dass etwas nicht stimmt. Gefährlich sind Programme, die fehlerfrei durchlaufen, einen Status OPTIMAL melden und eine Zah"},{"seite":"einfuehrung.html","titel":"1.11 Micro-Quiz","anker":"sec:einfuehrung-quiz","text":""},{"seite":"einfuehrung.html","titel":"1.12 Selbsttest","anker":"sec:einfuehrung-selbsttest","text":""},{"seite":"einfuehrung.html","titel":"1.13 Zusammenfassung","anker":"sec:einfuehrung-zusammenfassung","text":""},{"seite":"fundament.html","titel":"Kapitel 2: Das mathematische Fundament — Vektoren, Matrizen, Konvexität","anker":null,"text":""},{"seite":"fundament.html","titel":"2.1 In 5 Minuten gelöst","anker":"sec:fundament-schnellstart","text":"Dieses Kapitel handelt von Geometrie. Bevor wir Vektoren und Matrizen einführen, sehen Sie die zentrale Einsicht in Aktion — ganz ohne Solver, mit acht Zeilen NumPy."},{"seite":"fundament.html","titel":"2.2 Lernziele","anker":"sec:fundament-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"fundament.html","titel":"2.3 Warum überhaupt Vektoren und Matrizen?","anker":"sec:fundament-warum-ueberhaupt-vektoren-und-matrizen","text":"In Kapitel 1 hatten wir zwei Variablen und drei Nebenbedingungen. Das ließ sich bequem ausschreiben. Reale Modelle haben hunderte bis hunderttausende Variablen — dort wäre Ausschre"},{"seite":"fundament.html","titel":"Die Bausteine","anker":"die-bausteine","text":"Sei n die Anzahl der Entscheidungsvariablen und m die Anzahl der Nebenbedingungen."},{"seite":"fundament.html","titel":"Die kanonische Standardform","anker":"die-kanonische-standardform","text":"Damit lässt sich jedes lineare Programm in drei Zeilen schreiben:"},{"seite":"fundament.html","titel":"Dasselbe in Python","anker":"dasselbe-in-python","text":""},{"seite":"fundament.html","titel":"2.4 Der zulässige Lösungsraum und das Polyeder","anker":"sec:fundament-der-zulaessige-loesungsraum-und-das-polyeder","text":"Jede lineare Ungleichung a_{i1}x_1 + a_{i2}x_2 \\le b_i definiert geometrisch eine Halbebene (im \\mathbb{R}^2) bzw. einen Halbraum (im \\mathbb{R}^n): Die zugehörige Gleichung ist ei"},{"seite":"fundament.html","titel":"Der Fundamentalsatz der linearen Optimierung","anker":"der-fundamentalsatz-der-linearen-optimierung","text":""},{"seite":"fundament.html","titel":"2.5 Konvexität: die Grenze zwischen leicht und schwer","anker":"sec:fundament-konvexitaet-die-grenze-zwischen-leicht-und","text":"Hier kommt die vielleicht wichtigste Einsicht des ganzen Buches. In der Optimierungstheorie verläuft die Trennlinie zwischen „zuverlässig lösbar“ und „im Allgemeinen hoffnungslos“ "},{"seite":"fundament.html","titel":"Konvexe Menge","anker":"konvexe-menge","text":"Eine Menge \\mathcal{C} \\subseteq \\mathbb{R}^n heißt konvex, wenn für alle Punkte \\mathbf{x}, \\mathbf{y} \\in \\mathcal{C} und jedes \\theta \\in [0,1] gilt:"},{"seite":"fundament.html","titel":"Konvexe Funktion","anker":"konvexe-funktion","text":"Eine Funktion f: \\mathcal{C} \\to \\mathbb{R} heißt konvex, wenn für alle \\mathbf{x}, \\mathbf{y} \\in \\mathcal{C} und \\theta \\in [0,1] gilt:"},{"seite":"fundament.html","titel":"Der zentrale Satz","anker":"der-zentrale-satz","text":""},{"seite":"fundament.html","titel":"Was ist konvex, was nicht?","anker":"was-ist-konvex-was-nicht","text":""},{"seite":"fundament.html","titel":"Konvexität sichtbar machen","anker":"konvexität-sichtbar-machen","text":""},{"seite":"fundament.html","titel":"2.6 Geometrische Visualisierung des Lösungsraums","anker":"sec:fundament-geometrische-visualisierung-des-loesungsraums","text":"Wir zeichnen nun den Lösungsraum aus Kapitel 1 und lesen das Optimum ab. Zweidimensionale Bilder sind der schnellste Weg, Intuition für höhere Dimensionen aufzubauen — auch wenn ma"},{"seite":"fundament.html","titel":"2.7 Wenn die Zahlen nicht zusammenpassen: Kondition und Skalierung","anker":"sec:fundament-kondition","text":"Bis hierher war die lineare Algebra exakt. Ein Rechner rechnet aber nicht exakt, sondern mit rund 16 signifikanten Stellen — und Ihre Betriebsdaten sind ohnehin nur auf wenige Stel"},{"seite":"fundament.html","titel":"Die geometrische Vorstellung: zwei fast parallele Linien","anker":"die-geometrische-vorstellung-zwei-fast-parallele-linien","text":"Zwei Nebenbedingungen legen im Zweidimensionalen einen Schnittpunkt fest. Stehen die zugehörigen Geraden kreuzweise aufeinander, ist der Schnittpunkt robust: Verschiebt man eine Ge"},{"seite":"fundament.html","titel":"Woher schlechte Kondition im Alltag kommt","anker":"woher-schlechte-kondition-im-alltag-kommt","text":"Fast nie aus exotischer Mathematik — fast immer aus Einheiten:"},{"seite":"fundament.html","titel":"Ruiz-Equilibrierung: das Modell gesundrechnen","anker":"ruiz-equilibrierung-das-modell-gesundrechnen","text":"Die Gegenmaßnahme ist erstaunlich schlicht. Man multipliziert jede Zeile und jede Spalte der Matrix mit einem Faktor, sodass die Beträge überall in derselben Größenordnung landen:"},{"seite":"fundament.html","titel":"Das Experiment","anker":"das-experiment","text":"Das folgende Programm führt beides vor — die Fehlerverstärkung und ihre Behebung — und schließt mit der praktischen Konsequenz, die Sie in Kapitel 6 wieder brauchen werden: warum e"},{"seite":"fundament.html","titel":"2.8 Übungsaufgaben","anker":"sec:fundament-uebungsaufgaben","text":""},{"seite":"fundament.html","titel":"2.9 Finde den Denkfehler","anker":"sec:fundament-denkfehler","text":""},{"seite":"fundament.html","titel":"2.10 Micro-Quiz","anker":"sec:fundament-quiz","text":""},{"seite":"fundament.html","titel":"2.11 Selbsttest","anker":"sec:fundament-selbsttest","text":""},{"seite":"fundament.html","titel":"2.12 Zusammenfassung","anker":"sec:fundament-zusammenfassung","text":""},{"seite":"oekosystem.html","titel":"Kapitel 3: Das Python-Ökosystem für OR — Solver, Bindings und Modellierungsschichten","anker":null,"text":""},{"seite":"oekosystem.html","titel":"3.1 In 5 Minuten gelöst","anker":"sec:oekosystem-schnellstart","text":""},{"seite":"oekosystem.html","titel":"3.2 Lernziele","anker":"sec:oekosystem-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"oekosystem.html","titel":"3.3 Die Zwei-Schichten-Architektur","anker":"sec:oekosystem-die-zwei-schichten-architektur","text":"Im modernen Operations Research schreibt man Optimierungsalgorithmen nicht selbst. Man nutzt eine zweischichtige Architektur:"},{"seite":"oekosystem.html","titel":"3.4 Die Werkzeuge im Vergleich","anker":"sec:oekosystem-die-werkzeuge-im-vergleich","text":""},{"seite":"oekosystem.html","titel":"Die Entscheidung in drei Fragen","anker":"die-entscheidung-in-drei-fragen","text":"Statt die Tabelle auswendig zu lernen, beantworten Sie drei Fragen:"},{"seite":"oekosystem.html","titel":"3.5 Ein System — vier Programmieransätze","anker":"sec:oekosystem-ein-system-vier-programmieransaetze","text":"Nichts macht die Unterschiede so deutlich wie dasselbe Problem, viermal gelöst. Wir nehmen:"},{"seite":"oekosystem.html","titel":"Wie die Isolation aussieht, wenn sie tragen soll","anker":"wie-die-isolation-aussieht-wenn-sie-tragen-soll","text":"„Eigener Prozess” ist schnell gesagt. Die naheliegende Umsetzung — ein Codeschnipsel als Zeichenkette an python -c übergeben — funktioniert und ist trotzdem die schlechteste: Der S"},{"seite":"oekosystem.html","titel":"Was das CSR-Format bedeutet","anker":"was-das-csr-format-bedeutet","text":"highspy erwartet die Nebenbedingungsmatrix im CSR-Format (Compressed Sparse Row, komprimierte Zeilendarstellung). Statt der vollen Matrix speichert man nur die Einträge ungleich nu"},{"seite":"oekosystem.html","titel":"3.6 Wann lohnt sich welche Ebene?","anker":"sec:oekosystem-wann-lohnt-sich-welche-ebene","text":""},{"seite":"oekosystem.html","titel":"3.7 Modellierungsschichten für große Modelle: Pyomo und Linopy","anker":"sec:oekosystem-pyomo-linopy","text":"Die vier Bibliotheken aus dem Vierfach-Vergleich decken den Alltag weitgehend ab. Sobald Modelle industrielle Größe erreichen — zehntausende Variablen, Daten aus mehreren Systemen,"},{"seite":"oekosystem.html","titel":"Pyomo: die algebraische Denkweise","anker":"pyomo-die-algebraische-denkweise","text":"Pyomo ist im deutschsprachigen Raum der De-facto-Standard für große LP- und MILP-Modelle in Energiewirtschaft, Chemie und Logistik. Sein Kennzeichen: Es denkt in Mengen und Indizes"},{"seite":"oekosystem.html","titel":"Linopy: eine Zeile, zehntausend Nebenbedingungen","anker":"linopy-eine-zeile-zehntausend-nebenbedingungen","text":"Linopy verfolgt einen anderen Ansatz: Variablen sind beschriftete Arrays (xarray), keine indizierten Einzelobjekte. Damit wird aus"},{"seite":"oekosystem.html","titel":"Die Schichten im Überblick","anker":"die-schichten-im-überblick","text":""},{"seite":"oekosystem.html","titel":"3.8 Wo die Zeit wirklich hingeht: vektorisierte Modellgenerierung","anker":"sec:oekosystem-vektorisierung","text":"Eine der hartnäckigsten Fehlannahmen in Optimierungsprojekten lautet: „Wenn es zu langsam ist, brauchen wir einen besseren Solver.“ Messen Sie erst — oft stimmt das nicht."},{"seite":"oekosystem.html","titel":"Vier Stufen an einem Transportproblem","anker":"vier-stufen-an-einem-transportproblem","text":"Wir bauen dasselbe Transportproblem (m Werke, n Kunden, m \\cdot n Variablen) auf vier Arten auf:"},{"seite":"oekosystem.html","titel":"3.9 Übungsaufgaben","anker":"sec:oekosystem-uebungsaufgaben","text":""},{"seite":"oekosystem.html","titel":"3.10 Finde den Denkfehler","anker":"sec:oekosystem-denkfehler","text":""},{"seite":"oekosystem.html","titel":"3.11 Micro-Quiz","anker":"sec:oekosystem-quiz","text":""},{"seite":"oekosystem.html","titel":"3.12 Selbsttest","anker":"sec:oekosystem-selbsttest","text":""},{"seite":"oekosystem.html","titel":"3.13 Zusammenfassung","anker":"sec:oekosystem-zusammenfassung","text":""},{"seite":"modellierung.html","titel":"Kapitel 4: Vom Management-Wunsch zum Modell","anker":null,"text":""},{"seite":"modellierung.html","titel":"4.1 In 5 Minuten gelöst","anker":"sec:modellierung-schnellstart","text":""},{"seite":"modellierung.html","titel":"4.2 Lernziele","anker":"sec:modellierung-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"modellierung.html","titel":"4.3 Ein Satz ist noch kein Modell","anker":"sec:modellierung-vier-fragen","text":"Was aus einer Besprechung mitkommt, klingt meist so:"},{"seite":"modellierung.html","titel":"4.4 Derselbe Datensatz, fünf Modelle","anker":"sec:modellierung-fuenf-modelle","text":"Das Programm dieses Kapitels arbeitet die vier Fragen an einer Auftragsannahme ab: 14 Anfragen verlangen 800 Maschinenstunden, verfügbar sind 300."},{"seite":"modellierung.html","titel":"Teil 2: Ein Satz, drei Lesarten","anker":"teil-2-ein-satz-drei-lesarten","text":"„Stammkunden dürfen wir nicht verlieren.” Jeder im Raum nickt. Niemand merkt, dass drei verschiedene Modelle gemeint sein können:"},{"seite":"modellierung.html","titel":"Teil 3: Hart oder weich","anker":"teil-3-hart-oder-weich","text":"Dieselbe Regel als Strafkosten je abgelehntem Stammkundenauftrag:"},{"seite":"modellierung.html","titel":"Teil 4: Ein Quartal später","anker":"teil-4-ein-quartal-später","text":"Die harte Lesart A wurde eingebaut, alles lief drei Monate lang. Dann fragt Weber einen Auftrag über 320 Maschinenstunden an — mehr, als der ganze Zeitraum hergibt."},{"seite":"modellierung.html","titel":"4.5 Der Gesprächsleitfaden","anker":"sec:modellierung-leitfaden","text":"Was aus den vier Teilen als Handwerkszeug bleibt — Fragen, die man in der ersten Besprechung stellt, bevor eine Zeile Code entsteht:"},{"seite":"modellierung.html","titel":"4.6 Deutsch — OR: ein Übersetzungslexikon","anker":"sec:modellierung-glossar","text":"Die zwölf Fragen liefern Sätze. Sätze sind noch kein Modell — aber die meisten von ihnen haben eine Standardübersetzung, und wer sie kennt, spart sich das Nachdenken für die Fälle,"},{"seite":"modellierung.html","titel":"Fünf Wendungen, bei denen die Übersetzung eine Entscheidung erzwingt","anker":"fünf-wendungen-bei-denen-die-übersetzung-eine-entscheidung-erzwingt","text":"Die Tabelle oben tut so, als sei die Zuordnung eindeutig. Bei den folgenden fünf ist sie es nicht — und das Nachfragen ist die eigentliche Modellierungsarbeit:"},{"seite":"modellierung.html","titel":"Und die Gegenrichtung: OR — Deutsch","anker":"und-die-gegenrichtung-or-deutsch","text":"Dieselbe Barriere gibt es rückwärts. Diese Sätze haben sich bewährt, wenn ein Ergebnis jemandem erklärt werden muss, der das Modell nicht kennt:"},{"seite":"modellierung.html","titel":"4.7 Übungsaufgaben","anker":"sec:modellierung-uebungsaufgaben","text":""},{"seite":"modellierung.html","titel":"4.8 Finde den Denkfehler","anker":"sec:modellierung-denkfehler","text":""},{"seite":"modellierung.html","titel":"4.9 Micro-Quiz","anker":"sec:modellierung-quiz","text":""},{"seite":"modellierung.html","titel":"4.10 Selbsttest","anker":"sec:modellierung-selbsttest","text":""},{"seite":"modellierung.html","titel":"4.11 Zusammenfassung","anker":"sec:modellierung-zusammenfassung","text":""},{"seite":"synthese-grundlagen.html","titel":"Synthese Teil I","anker":null,"text":""},{"seite":"synthese-grundlagen.html","titel":"Was welches Kapitel klärt","anker":"was-welches-kapitel-klärt","text":""},{"seite":"synthese-grundlagen.html","titel":"Die Reihenfolge ist die Aussage","anker":"die-reihenfolge-ist-die-aussage","text":"Dieser Teil steht bewusst vor den Verfahren. Die drei häufigsten Gründe, aus denen ein OR-Projekt scheitert, liegen alle hier — und keiner davon ist ein Solverproblem:"},{"seite":"synthese-grundlagen.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"lp.html","titel":"Kapitel 5: Lineare Programmierung — Simplex, Dualität und Schattenpreise","anker":null,"text":""},{"seite":"lp.html","titel":"5.1 In 5 Minuten gelöst","anker":"sec:lp-schnellstart","text":"Bisher hat ein Solver Ihnen gesagt, was Sie tun sollen. Jetzt sagt er Ihnen zusätzlich, was eine zusätzliche Stunde wert wäre — und das ist die Zahl, für die Sie im Betrieb gehört "},{"seite":"lp.html","titel":"5.2 Lernziele","anker":"sec:lp-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"lp.html","titel":"5.3 Die Standardform und Schlupfvariablen","anker":"sec:lp-die-standardform-und-schlupfvariablen","text":"Lineare Programme zeichnen sich dadurch aus, dass Zielfunktion und alle Nebenbedingungen strikt linear sind — keine Produkte von Variablen, keine Quadrate, keine Logarithmen."},{"seite":"lp.html","titel":"5.4 Der Simplex-Algorithmus Schritt für Schritt","anker":"sec:lp-der-simplex-algorithmus-schritt-fuer-schritt","text":"Der von George Dantzig 1947 formulierte Simplex-Algorithmus nutzt den Fundamentalsatz aus Kapitel 2: Das Optimum liegt in einer Ecke. Statt alle Ecken aufzuzählen, wandert er von E"},{"seite":"lp.html","titel":"Der Ablauf","anker":"der-ablauf","text":""},{"seite":"lp.html","titel":"5.5 Das Simplex-Tableau in Python","anker":"sec:lp-das-simplex-tableau-in-python","text":""},{"seite":"lp.html","titel":"5.6 Dualität und Schattenpreise","anker":"sec:lp-dualitaet-und-schattenpreise","text":"Zu jedem linearen Optimierungsproblem (dem primalen Problem) existiert ein spiegelbildliches Zwillingsproblem: das duale Problem."},{"seite":"lp.html","titel":"Der starke Dualitätssatz","anker":"der-starke-dualitätssatz","text":""},{"seite":"lp.html","titel":"Der Schattenpreis als Ableitung","anker":"der-schattenpreis-als-ableitung","text":"Der Dualwert y_i^* ist die Grenzproduktivität der Ressource i:"},{"seite":"lp.html","titel":"5.7 Die Vorzeichenfalle bei Schattenpreisen","anker":"sec:lp-die-vorzeichenfalle-bei-schattenpreisen","text":"Diese Falle sollten Sie sich merken, weil sie in jedem Projekt wieder auftaucht."},{"seite":"lp.html","titel":"5.8 Praxisfall: Sensitivitätsanalyse mit korrekten Schattenpreisen","anker":"sec:lp-praxisfall-sensitivitaetsanalyse-mit-korrekten-schattenpreisen","text":"Szenario. Ein Fertigungsbetrieb stellt drei Produkte her und will wissen:"},{"seite":"lp.html","titel":"5.9 Wann Schattenpreise lügen: Entartung und Toleranzen","anker":"sec:lp-entartung","text":"Der vorige Abschnitt hat gezeigt, wie wertvoll Schattenpreise sind. Dieser zeigt die zwei Fälle, in denen sie eine Entscheidung nicht tragen — und wie man beide erkennt, bevor jema"},{"seite":"lp.html","titel":"Fall 1: Entartung","anker":"fall-1-entartung","text":"Erinnern Sie sich an Abschnitt 2.4: In zwei Dimensionen legen zwei sich schneidende Geraden eine Ecke fest, in drei Dimensionen drei Ebenen, allgemein n Restriktionen bei n Variabl"},{"seite":"lp.html","titel":"Fall 2: Toleranzen","anker":"fall-2-toleranzen","text":"Der zweite Fall ist unscheinbarer und deshalb häufiger. Wie stellt man fest, ob eine Ressource ausgelastet ist? Naheliegend wäre if schlupf == 0. Genau das ist falsch."},{"seite":"lp.html","titel":"Die ehrliche Auskunft: Schattenpreis-Spannen","anker":"die-ehrliche-auskunft-schattenpreis-spannen","text":"Was tut man also, wenn das Modell entartet ist? Man meldet keinen Einzelwert, sondern die Spanne. Die Menge aller optimalen Dualvektoren ist selbst ein Polyeder:"},{"seite":"lp.html","titel":"5.10 Übungsaufgaben","anker":"sec:lp-uebungsaufgaben","text":""},{"seite":"lp.html","titel":"5.11 Finde den Denkfehler","anker":"sec:lp-denkfehler","text":""},{"seite":"lp.html","titel":"5.12 Micro-Quiz","anker":"sec:lp-quiz","text":""},{"seite":"lp.html","titel":"5.13 Selbsttest","anker":"sec:lp-selbsttest","text":""},{"seite":"lp.html","titel":"5.14 Zusammenfassung","anker":"sec:lp-zusammenfassung","text":""},{"seite":"milp.html","titel":"Kapitel 6: Gemischt-ganzzahlige Optimierung — Diskrete Entscheidungen und Branch-and-Bound","anker":null,"text":""},{"seite":"milp.html","titel":"6.1 In 5 Minuten gelöst","anker":"sec:milp-schnellstart","text":""},{"seite":"milp.html","titel":"6.2 Lernziele","anker":"sec:milp-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"milp.html","titel":"6.3 Warum Runden fundamental scheitert","anker":"sec:milp-warum-runden-fundamental-scheitert","text":"In der realen Welt sind viele Entscheidungen nicht teilbar. Man kann nicht 0,47 Flugzeuge kaufen, keine halbe Lagerhalle bauen, keinen Mitarbeiter zu 38 % einstellen. Noch wichtige"},{"seite":"milp.html","titel":"6.4 Branch-and-Bound","anker":"sec:milp-branch-and-bound","text":"MILP-Probleme sind NP-schwer. Der Standardansatz moderner Solver (HiGHS, SCIP, Gurobi) heißt Branch-and-Cut — Branch-and-Bound plus Schnittebenen."},{"seite":"milp.html","titel":"Die drei Phasen","anker":"die-drei-phasen","text":""},{"seite":"milp.html","titel":"6.5 Modellierungstricks: Big-M und logische Bedingungen","anker":"sec:milp-modellierungstricks-big-m-und-logische-bedingungen","text":"Hier kommt das Handwerkszeug, das MILP so mächtig macht: Man übersetzt Geschäftsregeln in lineare Ungleichungen — mithilfe binärer Hilfsvariablen y \\in \\{0,1\\} und einer hinreichen"},{"seite":"milp.html","titel":"Muster 1 — Fixkosten / Aktivierungsschalter","anker":"muster-1-fixkosten-aktivierungsschalter","text":"Regel: Wird Maschine j genutzt (x_j &gt; 0), fallen Rüstkosten F_j an."},{"seite":"milp.html","titel":"Muster 2 — Entweder-Oder (disjunctive constraints)","anker":"muster-2-entweder-oder-disjunctive-constraints","text":"Regel: Es muss entweder f(\\mathbf{x}) \\le b_1 oder g(\\mathbf{x}) \\le b_2 gelten."},{"seite":"milp.html","titel":"Muster 3 — Wenn-Dann (Implikation)","anker":"muster-3-wenn-dann-implikation","text":"Regel: Wenn y_1 = 1, dann muss auch y_2 = 1 sein."},{"seite":"milp.html","titel":"Muster 4 — Kardinalität („höchstens K aus N“)","anker":"muster-4-kardinalität-höchstens-k-aus-n","text":"\\sum_{j=1}^N y_j \\le K"},{"seite":"milp.html","titel":"Muster 5 — Semikontinuierlich („entweder 0 oder mindestens L“)","anker":"muster-5-semikontinuierlich-entweder-0-oder-mindestens-l","text":"Regel: Eine Position ist entweder gar nicht besetzt oder mit mindestens L Euro."},{"seite":"milp.html","titel":"6.6 Beispiel: Das Rucksackproblem","anker":"sec:milp-beispiel-das-rucksackproblem","text":"Bevor wir zum Portfolio kommen, das klassische Einstiegsproblem — kurz, verständlich und überall wiederzuerkennen."},{"seite":"milp.html","titel":"6.7 Praxisfall: Portfolio mit Ordergebühren und Kardinalitätsgrenze","anker":"sec:milp-praxisfall-portfolio-mit-ordergebuehren-und-kardinalitaetsgrenze","text":"Szenario. Ein Investor verteilt 100\\,000\\,\\text{€} auf sechs Anlageklassen."},{"seite":"milp.html","titel":"6.8 Wenn der Solver nicht fertig wird: Gap, Zeitlimit und Warm-Start","anker":"sec:milp-gap","text":"Bei einem LP gibt es zwei Ausgänge: eine optimale Lösung oder eine klare Absage. Bei einem MILP gibt es einen dritten, und im Betrieb ist er der häufigste:"},{"seite":"milp.html","titel":"Die zwei Zahlen, die zählen","anker":"die-zwei-zahlen-die-zählen","text":"Branch-and-Bound führt zu jedem Zeitpunkt zwei Werte mit:"},{"seite":"milp.html","titel":"Alle Statusfälle behandeln","anker":"alle-statusfälle-behandeln","text":"Der zweite Teil der Professionalität ist unspektakulär, aber entscheidend: jeden Rückgabewert auswerten, nicht nur OPTIMAL."},{"seite":"milp.html","titel":"Warm-Starts: was sie können und was nicht","anker":"warm-starts-was-sie-können-und-was-nicht","text":"Ein Warm-Start gibt dem Solver eine bekannte Lösung als Startpunkt mit. Die Idee ist verlockend: Wer schon eine brauchbare Lösung hat, muss nicht bei null anfangen."},{"seite":"milp.html","titel":"6.9 Übungsaufgaben","anker":"sec:milp-uebungsaufgaben","text":""},{"seite":"milp.html","titel":"6.10 Finde den Denkfehler","anker":"sec:milp-denkfehler","text":"Die Warnung „M so klein wie möglich“ wird meist mit der Laufzeit begründet: schwache Relaxation, mehr Knoten. Das stimmt — und ist die harmlosere Hälfte der Wahrheit."},{"seite":"milp.html","titel":"6.11 Micro-Quiz","anker":"sec:milp-quiz","text":""},{"seite":"milp.html","titel":"6.12 Selbsttest","anker":"sec:milp-selbsttest","text":""},{"seite":"milp.html","titel":"6.13 Zusammenfassung","anker":"sec:milp-zusammenfassung","text":""},{"seite":"cpsat.html","titel":"Kapitel 7: Constraint Programming mit CP-SAT — Logik, Scheduling und Zuweisung","anker":null,"text":""},{"seite":"cpsat.html","titel":"7.1 In 5 Minuten gelöst","anker":"sec:cpsat-schnellstart","text":""},{"seite":"cpsat.html","titel":"7.2 Lernziele","anker":"sec:cpsat-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"cpsat.html","titel":"7.3 Ein anderes Denkmodell","anker":"sec:cpsat-ein-anderes-denkmodell","text":"Während LP und MILP auf algebraischen Gleichungen und Simplex-Matrizen beruhen, kommt Constraint Programming (CP) aus der künstlichen Intelligenz:"},{"seite":"cpsat.html","titel":"Was Propagation leistet","anker":"was-propagation-leistet","text":""},{"seite":"cpsat.html","titel":"7.4 Globale Constraints — die Bausteine von CP-SAT","anker":"sec:cpsat-globale-constraints-die-bausteine-von-cp","text":"CP-SAT stellt mächtige Makros bereit, die in MILP hunderte Big-M-Ungleichungen erfordern würden. Jedes davon hat einen spezialisierten, hocheffizienten Propagator."},{"seite":"cpsat.html","titel":"7.5 Praxisfall: Dynamisches Vertretungssystem","anker":"sec:cpsat-praxisfall-dynamisches-vertretungssystem","text":"Problemstellung. Eine Schule muss nach spontanen Ausfällen vier Unterrichtsstunden nachbesetzen."},{"seite":"cpsat.html","titel":"7.6 Intervallvariablen: Job-Shop-Scheduling","anker":"sec:cpsat-intervallvariablen-job-shop-scheduling","text":"Die Königsdisziplin von CP-SAT sind Zeitplanungsprobleme. Intervallvariablen sind dabei das zentrale Werkzeug."},{"seite":"cpsat.html","titel":"7.7 Parallele Suche: was num_workers wirklich bewirkt","anker":"sec:cpsat-parallele-suche","text":"An mehreren Stellen dieses Buchs steht loeser.parameters.num_workers = 1, mit dem Kommentar „für eine reproduzierbare Ausgabe“. Darin stecken zwei Behauptungen: dass parallele Such"},{"seite":"cpsat.html","titel":"Was die Messung zeigt","anker":"was-die-messung-zeigt","text":"Erstens: Die Beschleunigung ist überlinear. Acht Arbeiter sind im abgedruckten Lauf nicht achtmal, sondern gut zwölfmal schneller als einer. Das sieht nach Messfehler aus, ist aber"},{"seite":"cpsat.html","titel":"Was daraus für Tests folgt","anker":"was-daraus-für-tests-folgt","text":"Der Zielwert war in allen sechzehn Läufen derselbe: 183. Die Gestalt der Lösung war es nicht. Genau daran entscheidet sich, was ein Test prüfen darf:"},{"seite":"cpsat.html","titel":"7.8 Die fünf Antworten von CP-SAT","anker":"sec:cpsat-statusfaelle","text":"Abschnitt 6.8 hat gezeigt, warum OPTIMAL bei ganzzahligen Problemen nicht der Normalfall ist. Bei CP-SAT kommt eine Besonderheit hinzu: Es gibt fünf verschiedene Antworten, und dre"},{"seite":"cpsat.html","titel":"7.9 Übungsaufgaben","anker":"sec:cpsat-uebungsaufgaben","text":""},{"seite":"cpsat.html","titel":"7.10 Finde den Denkfehler","anker":"sec:cpsat-denkfehler","text":""},{"seite":"cpsat.html","titel":"7.11 Micro-Quiz","anker":"sec:cpsat-quiz","text":""},{"seite":"cpsat.html","titel":"7.12 Selbsttest","anker":"sec:cpsat-selbsttest","text":""},{"seite":"cpsat.html","titel":"7.13 Zusammenfassung","anker":"sec:cpsat-zusammenfassung","text":""},{"seite":"graphen.html","titel":"Kapitel 8: Graphen, Flüsse und Touren — Min-Cost-Flow, Matching und VRP","anker":null,"text":""},{"seite":"graphen.html","titel":"8.1 In 5 Minuten gelöst","anker":"sec:graphen-schnellstart","text":""},{"seite":"graphen.html","titel":"8.2 Lernziele","anker":"sec:graphen-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"graphen.html","titel":"8.3 Graphen als Modellsprache","anker":"sec:graphen-graphen-als-modellsprache","text":"Viele reale Probleme in Logistik, Kommunikation und Finanzströmen sind keine flachen Ungleichungssysteme, sondern Graphen G = (V, E):"},{"seite":"graphen.html","titel":"Das Minimum-Cost-Flow-Problem (MCNFP)","anker":"das-minimum-cost-flow-problem-mcnfp","text":"Minimum-Cost Network Flow Problem — deutsch: kostenminimales Flussproblem. Es ist das mathematische Fundament für Transportketten, Liquiditätsrouting und Datenverteilung."},{"seite":"graphen.html","titel":"8.4 Bipartites Matching: das Zuordnungsproblem","anker":"sec:graphen-bipartites-matching-das-zuordnungsproblem","text":"Wenn N Aufgaben auf N Ressourcen eins zu eins verteilt werden — Orders auf Broker, Schichten auf Mitarbeitende, Aufträge auf Maschinen — spricht man von bipartitem Matching."},{"seite":"graphen.html","titel":"Der Satz von Birkhoff und von Neumann","anker":"der-satz-von-birkhoff-und-von-neumann","text":""},{"seite":"graphen.html","titel":"8.5 Das Vehicle Routing Problem mit Zeitfenstern","anker":"sec:graphen-das-vehicle-routing-problem-mit-zeitfenstern","text":"Das Traveling Salesperson Problem (TSP), deutsch Problem des Handlungsreisenden, fragt nach der kürzesten Rundreise durch N Städte. Das Capacitated Vehicle Routing Problem with Tim"},{"seite":"graphen.html","titel":"Kurzzyklen verhindern","anker":"kurzzyklen-verhindern","text":"Ein naives Modell erlaubt Subtouren: isolierte Kreise, die das Depot nie anfahren. Die klassische Gegenmaßnahme ist die MTZ-Formulierung nach Miller, Tucker und Zemlin. Man führt R"},{"seite":"graphen.html","titel":"Praxisbeispiel: Flotten-Routing","anker":"praxisbeispiel-flotten-routing","text":""},{"seite":"graphen.html","titel":"8.6 Übungsaufgaben","anker":"sec:graphen-uebungsaufgaben","text":""},{"seite":"graphen.html","titel":"8.7 Finde den Denkfehler","anker":"sec:graphen-denkfehler","text":""},{"seite":"graphen.html","titel":"8.8 Micro-Quiz","anker":"sec:graphen-quiz","text":""},{"seite":"graphen.html","titel":"8.9 Selbsttest","anker":"sec:graphen-selbsttest","text":""},{"seite":"graphen.html","titel":"8.10 Zusammenfassung","anker":"sec:graphen-zusammenfassung","text":""},{"seite":"metaheuristiken.html","titel":"Kapitel 9: Metaheuristiken — wenn der exakte Solver aussteigt","anker":null,"text":""},{"seite":"metaheuristiken.html","titel":"9.1 In 5 Minuten gelöst","anker":"sec:metaheuristiken-schnellstart","text":""},{"seite":"metaheuristiken.html","titel":"9.2 Lernziele","anker":"sec:metaheuristiken-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"metaheuristiken.html","titel":"9.3 Die Aufgabe: Rüstzeiten an der Lackieranlage","anker":"sec:metaheuristiken-aufgabe","text":"Der Schnellstart war ein Spielzeug. Die Aufgabe dieses Kapitels ist dieselbe, nur in Betriebsgröße: n Aufträge in eine Reihenfolge bringen, so dass die Summe der Rüstzeiten minimal"},{"seite":"metaheuristiken.html","titel":"Die Instanz","anker":"die-instanz","text":"Alle Programme dieses Kapitels benutzen dieselbe Aufgabe. Die Rüstzeit setzt sich aus zwei Teilen zusammen: einem unregelmäßigen Reinigungsaufwand zwischen den Produktfamilien und "},{"seite":"metaheuristiken.html","titel":"9.4 Lokale Suche: der Zug und seine Kosten","anker":"sec:metaheuristiken-lokale-suche","text":"Eine lokale Suche braucht drei Dinge: eine Startlösung, einen Zug und eine Regel, welche Züge übernommen werden."},{"seite":"metaheuristiken.html","titel":"9.5 Simulated Annealing","anker":"sec:metaheuristiken-annealing","text":"Reines Bergsteigen — nur Verbesserungen annehmen — bleibt im ersten lokalen Optimum stehen. Simulated Annealing nimmt Verschlechterungen mit einer Wahrscheinlichkeit an, die von de"},{"seite":"metaheuristiken.html","titel":"Was die Messung zeigt — und was sie nicht zeigt","anker":"was-die-messung-zeigt-und-was-sie-nicht-zeigt","text":"Erstens: Der Beitrag des Annealings ist bescheiden. Bergsteigen kommt auf 917 Minuten, Annealing auf 907 — das simple Verfahren holt 93 % des Ertrags. Das ist kein Argument gegen A"},{"seite":"metaheuristiken.html","titel":"9.6 Ab wann lohnt es sich?","anker":"sec:metaheuristiken-umschlagpunkt","text":"„Kleine Probleme exakt, große mit Metaheuristiken“ ist richtig und nutzlos, solange niemand sagt, wo groß anfängt. Diese Zahl lässt sich messen."},{"seite":"metaheuristiken.html","titel":"Die Spalte, die man nicht weglassen darf","anker":"die-spalte-die-man-nicht-weglassen-darf","text":"Bei 500 Aufträgen haben wir die CP-SAT-Lösung verworfen. Seine Schranke ist trotzdem das Wertvollste in der ganzen Tabelle."},{"seite":"metaheuristiken.html","titel":"9.7 Large Neighborhood Search","anker":"sec:metaheuristiken-lns","text":"Die Tabelle oben endet mit einem Widerspruch. Bei 500 Aufträgen scheitert der exakte Solver an der Größe; die Metaheuristik scheitert an der Kleinheit ihrer Züge — sie verschiebt j"},{"seite":"metaheuristiken.html","titel":"Das Fenster ist die eine Stellschraube","anker":"das-fenster-ist-die-eine-stellschraube","text":""},{"seite":"metaheuristiken.html","titel":"Das Gesamtbild","anker":"das-gesamtbild","text":""},{"seite":"metaheuristiken.html","titel":"9.8 Was dieses Kapitel nicht behandelt","anker":"sec:metaheuristiken-abgrenzung","text":"Damit die Landkarte vollständig ist — drei Verfahren, die in dieselbe Familie gehören und hier bewusst nur benannt werden:"},{"seite":"metaheuristiken.html","titel":"9.9 Übungsaufgaben","anker":"sec:metaheuristiken-uebungsaufgaben","text":""},{"seite":"metaheuristiken.html","titel":"9.10 Finde den Denkfehler","anker":"sec:metaheuristiken-denkfehler","text":""},{"seite":"metaheuristiken.html","titel":"9.11 Micro-Quiz","anker":"sec:metaheuristiken-quiz","text":""},{"seite":"metaheuristiken.html","titel":"9.12 Selbsttest","anker":"sec:metaheuristiken-selbsttest","text":""},{"seite":"metaheuristiken.html","titel":"9.13 Zusammenfassung","anker":"sec:metaheuristiken-zusammenfassung","text":""},{"seite":"dekomposition.html","titel":"Kapitel 10: Spaltengenerierung — das Modell umbauen statt die Lösung raten","anker":null,"text":""},{"seite":"dekomposition.html","titel":"10.1 In 5 Minuten gelöst","anker":"sec:dekomposition-schnellstart","text":""},{"seite":"dekomposition.html","titel":"10.2 Lernziele","anker":"sec:dekomposition-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"dekomposition.html","titel":"10.3 Warum das naheliegende Modell scheitert","anker":"sec:dekomposition-symmetrie","text":"Der erste Entwurf eines Zuschnittmodells sieht fast immer so aus: eine Binärvariable z_{jr} für „Stück j liegt auf Rolle r“, dazu y_r für „Rolle r wird benutzt”."},{"seite":"dekomposition.html","titel":"10.4 Rechnen mit Mustern, ohne sie aufzuschreiben","anker":"sec:dekomposition-schleife","text":"Die Idee ist alt (Gilmore und Gomory, 1961) und in einem Satz gesagt:"},{"seite":"dekomposition.html","titel":"10.5 Das Programm","anker":"sec:dekomposition-programm","text":""},{"seite":"dekomposition.html","titel":"10.6 Was die Zahlen zeigen","anker":"sec:dekomposition-befund","text":""},{"seite":"dekomposition.html","titel":"Die Musterzahl explodiert, der Bedarf nicht","anker":"die-musterzahl-explodiert-der-bedarf-nicht","text":""},{"seite":"dekomposition.html","titel":"Der unbequeme Teil: Es lohnt sich nicht immer","anker":"der-unbequeme-teil-es-lohnt-sich-nicht-immer","text":""},{"seite":"dekomposition.html","titel":"10.7 Übungsaufgaben","anker":"sec:dekomposition-uebungsaufgaben","text":""},{"seite":"dekomposition.html","titel":"10.8 Finde den Denkfehler","anker":"sec:dekomposition-denkfehler","text":""},{"seite":"dekomposition.html","titel":"10.9 Micro-Quiz","anker":"sec:dekomposition-quiz","text":""},{"seite":"dekomposition.html","titel":"10.10 Selbsttest","anker":"sec:dekomposition-selbsttest","text":""},{"seite":"dekomposition.html","titel":"10.11 Zusammenfassung","anker":"sec:dekomposition-zusammenfassung","text":""},{"seite":"synthese-kernverfahren.html","titel":"Synthese Teil II","anker":null,"text":""},{"seite":"synthese-kernverfahren.html","titel":"Die Entscheidungsmatrix","anker":"die-entscheidungsmatrix","text":""},{"seite":"synthese-kernverfahren.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-kernverfahren.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-kernverfahren.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"qp-nlp.html","titel":"Kapitel 11: Quadratische und nichtlineare Optimierung — KKT, Lagrange, Konvexität","anker":null,"text":""},{"seite":"qp-nlp.html","titel":"11.1 In 5 Minuten gelöst","anker":"sec:qp-nlp-schnellstart","text":""},{"seite":"qp-nlp.html","titel":"11.2 Lernziele","anker":"sec:qp-nlp-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"qp-nlp.html","titel":"Auffrischung: Gradient in einer Minute","anker":"auffrischung-gradient-in-einer-minute","text":"Der Gradient \\nabla f(\\mathbf{x}) ist der Vektor aller partiellen Ableitungen:"},{"seite":"qp-nlp.html","titel":"11.3 Das quadratische Programm (QP)","anker":"sec:qp-nlp-das-quadratische-programm-qp","text":"Ein quadratisches Programm hat eine quadratische Zielfunktion und lineare Nebenbedingungen:"},{"seite":"qp-nlp.html","titel":"Konvexität: die präzise Aussage","anker":"konvexität-die-präzise-aussage","text":"Eine verbreitete, aber zu stark formulierte Aussage lautet: „Wenn \\mathbf{P} positiv semi-definit ist, ist das QP streng konvex; es existiert ein eindeutiges globales Minimum.“ Kor"},{"seite":"qp-nlp.html","titel":"11.4 Die Karush-Kuhn-Tucker-Bedingungen","anker":"sec:qp-nlp-die-karush-kuhn-tucker-bedingungen","text":"Die KKT-Bedingungen verallgemeinern den Lagrange-Ansatz auf Probleme mit Ungleichungen. Sie sind notwendig für ein Optimum — und bei konvexen Problemen auch hinreichend."},{"seite":"qp-nlp.html","titel":"Die vier KKT-Bedingungen","anker":"die-vier-kkt-bedingungen","text":"Am Optimum (\\mathbf{x}^*, \\boldsymbol{\\lambda}^*, \\boldsymbol{\\nu}^*) gilt:"},{"seite":"qp-nlp.html","titel":"11.5 Nichtlineare Optimierung mit scipy.optimize.minimize","anker":"sec:qp-nlp-nichtlineare-optimierung-mit-scipy-optimize-minimize","text":"Wenn Zielfunktion oder Nebenbedingungen weder linear noch quadratisch sind — Logarithmen, Wurzeln, Exponentialfunktionen — nutzt man gradientenbasierte Verfahren wie SLSQP (Sequent"},{"seite":"qp-nlp.html","titel":"Praxisfall: Entropie-maximierte Kapitalallokation","anker":"praxisfall-entropie-maximierte-kapitalallokation","text":"Wir optimieren ein Portfolio nicht nur nach Varianz, sondern maximieren zusätzlich die Shannon-Entropie der Gewichte, um Klumpenrisiken glatt zu bestrafen:"},{"seite":"qp-nlp.html","titel":"11.6 Jenseits der Konvexität: lokale Optima, Multistart und MINLP","anker":"sec:qp-nlp-nichtkonvex","text":"Alles bisher in diesem Kapitel stand unter einer Bedingung: Konvexität. Sie ist der Grund, warum CVXPY eine Optimalitätsgarantie geben kann und warum die KKT-Bedingungen nicht nur "},{"seite":"qp-nlp.html","titel":"Woher Nicht-Konvexität im Alltag kommt","anker":"woher-nicht-konvexität-im-alltag-kommt","text":"Nicht aus exotischer Mathematik, sondern aus ganz gewöhnlichen betriebswirtschaftlichen Effekten:"},{"seite":"qp-nlp.html","titel":"Was ein lokales Optimum praktisch bedeutet","anker":"was-ein-lokales-optimum-praktisch-bedeutet","text":"scipy.optimize.minimize verweigert nichts. Es rechnet, meldet success: True und liefert eine Zahl. Diese Meldung heißt aber nicht „das ist das Optimum“, sondern nur:"},{"seite":"qp-nlp.html","titel":"Die Zahlen im Klartext","anker":"die-zahlen-im-klartext","text":""},{"seite":"qp-nlp.html","titel":"Ausblick: MINLP und globale Solver","anker":"ausblick-minlp-und-globale-solver","text":"Wird zusätzlich noch ganzzahlig entschieden — „welche Anlage läuft überhaupt“ und „wie stark“ —, entsteht ein MINLP (Mixed-Integer Nonlinear Program), die schwierigste der in diese"},{"seite":"qp-nlp.html","titel":"11.7 Übungsaufgaben","anker":"sec:qp-nlp-uebungsaufgaben","text":""},{"seite":"qp-nlp.html","titel":"11.8 Finde den Denkfehler","anker":"sec:qp-nlp-denkfehler","text":""},{"seite":"qp-nlp.html","titel":"11.9 Micro-Quiz","anker":"sec:qp-nlp-quiz","text":""},{"seite":"qp-nlp.html","titel":"11.10 Selbsttest","anker":"sec:qp-nlp-selbsttest","text":""},{"seite":"qp-nlp.html","titel":"11.11 Zusammenfassung","anker":"sec:qp-nlp-zusammenfassung","text":""},{"seite":"unsicherheit.html","titel":"Kapitel 12: Optimierung unter Unsicherheit — Monte-Carlo, Stochastik, Robustheit","anker":null,"text":""},{"seite":"unsicherheit.html","titel":"12.1 In 5 Minuten gelöst","anker":"sec:unsicherheit-schnellstart","text":""},{"seite":"unsicherheit.html","titel":"12.2 Lernziele","anker":"sec:unsicherheit-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"unsicherheit.html","titel":"12.3 Der Fluch des Durchschnitts","anker":"sec:unsicherheit-der-fluch-des-durchschnitts","text":"Der naheliegende Umgang mit Unsicherheit lautet: „Wir setzen den Erwartungswert ein und rechnen deterministisch.“ Das ist fast immer falsch — und zwar nicht ungenau, sondern system"},{"seite":"unsicherheit.html","titel":"Die vier Ansätze im Überblick","anker":"die-vier-ansätze-im-überblick","text":""},{"seite":"unsicherheit.html","titel":"12.4 Monte-Carlo-Simulation","anker":"sec:unsicherheit-monte-carlo-simulation","text":"Monte-Carlo heißt schlicht: Ziehe sehr viele Zufallsszenarien, werte für jedes den Plan aus, und betrachte die Verteilung der Ergebnisse statt eines einzigen Werts."},{"seite":"unsicherheit.html","titel":"12.5 Zweistufige stochastische Programmierung","anker":"sec:unsicherheit-zweistufige-stochastische-programmierung","text":"Das zweistufige Modell mit Recourse bildet ab, dass Entscheidungen zeitlich gestaffelt sind:"},{"seite":"unsicherheit.html","titel":"12.6 Robuste Optimierung: gegen den Worst Case absichern","anker":"sec:unsicherheit-robuste-optimierung-gegen-den-worst-case","text":"Wenn Wahrscheinlichkeiten unbekannt oder instabil sind — Marktcrashs, Lieferkettenabrisse, Pandemien —, hilft die robuste Optimierung. Sie fragt nicht nach dem Mittel, sondern nach"},{"seite":"unsicherheit.html","titel":"12.7 Wahrscheinlichkeitsbeschränkungen: „mit 95 % Sicherheit“","anker":"sec:unsicherheit-chance-constraints","text":"Die drei bisherigen Ansätze beantworten Fragen, die im Sitzungssaal selten so gestellt werden. „Was ist im Mittel am besten?“ hört ein Vorstand als Ausflucht; „was hält auch im sch"},{"seite":"unsicherheit.html","titel":"Weg 1: Normalverteilung — die Bedingung wird ein Kegel","anker":"weg-1-normalverteilung-die-bedingung-wird-ein-kegel","text":"Ist \\mathbf{a} \\sim \\mathcal{N}(\\hat{\\mathbf{a}}, \\boldsymbol{\\Sigma}), dann ist \\mathbf{a}^\\top\\mathbf{x} für festes \\mathbf{x} selbst normalverteilt, mit Erwartungswert \\hat{\\mat"},{"seite":"unsicherheit.html","titel":"Weg 2: Szenarien — Big-M ohne Verteilungsannahme","anker":"weg-2-szenarien-big-m-ohne-verteilungsannahme","text":"Liegen statt einer Verteilung nur S beobachtete Szenarien vor, bekommt jedes eine Binärvariable z_s, die sagt, ob es verletzt werden darf:"},{"seite":"unsicherheit.html","titel":"Der Fall: ein Kraftwerkspark mit 500 MW Zusage","anker":"der-fall-ein-kraftwerkspark-mit-500-mw-zusage","text":""},{"seite":"unsicherheit.html","titel":"12.8 Übungsaufgaben","anker":"sec:unsicherheit-uebungsaufgaben","text":""},{"seite":"unsicherheit.html","titel":"12.9 Finde den Denkfehler","anker":"sec:unsicherheit-denkfehler","text":""},{"seite":"unsicherheit.html","titel":"12.10 Micro-Quiz","anker":"sec:unsicherheit-quiz","text":""},{"seite":"unsicherheit.html","titel":"12.11 Selbsttest","anker":"sec:unsicherheit-selbsttest","text":""},{"seite":"unsicherheit.html","titel":"12.12 Zusammenfassung","anker":"sec:unsicherheit-zusammenfassung","text":""},{"seite":"dynamische-programmierung.html","titel":"Kapitel 13: Dynamische Programmierung — Die Bellman-Gleichung und Order-Execution","anker":null,"text":""},{"seite":"dynamische-programmierung.html","titel":"13.1 In 5 Minuten gelöst","anker":"sec:dynamische-programmierung-schnellstart","text":""},{"seite":"dynamische-programmierung.html","titel":"13.2 Lernziele","anker":"sec:dynamische-programmierung-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"dynamische-programmierung.html","titel":"13.3 Das Bellmansche Optimalitätsprinzip","anker":"sec:dynamische-programmierung-das-bellmansche-optimalitaetsprinzip","text":"Richard Bellman formulierte 1957 das Grundprinzip der Dynamischen Programmierung (DP):"},{"seite":"dynamische-programmierung.html","titel":"Die vier Bausteine eines DP-Modells","anker":"die-vier-bausteine-eines-dp-modells","text":""},{"seite":"dynamische-programmierung.html","titel":"Die Bellman-Gleichung","anker":"die-bellman-gleichung","text":"V_t(S_t) = \\min_{a_t\\in\\mathcal{A}(S_t)}\\Big\\{\\,C(S_t,a_t) \\;+\\; \\gamma\\cdot\\mathbb{E}\\big[V_{t+1}(S_{t+1})\\mid S_t,a_t\\big]\\Big\\}"},{"seite":"dynamische-programmierung.html","titel":"13.4 Das Almgren-Chriss-Problem: optimale Orderausführung","anker":"sec:dynamische-programmierung-das-almgren-chriss-problem-optimale-orderausfuehrung","text":"Ein klassisches OR-Problem an Börsen: Ein institutioneller Händler muss X_0 = 100\\,000 Aktien innerhalb von T Handelsperioden verkaufen."},{"seite":"dynamische-programmierung.html","titel":"13.5 Der Fluch der Dimensionalität","anker":"sec:dynamische-programmierung-der-fluch-der-dimensionalitaet","text":"DP ist mächtig, aber es hat eine harte Grenze. Der Aufwand der Rückwärtsinduktion ist"},{"seite":"dynamische-programmierung.html","titel":"13.6 Übungsaufgaben","anker":"sec:dynamische-programmierung-uebungsaufgaben","text":""},{"seite":"dynamische-programmierung.html","titel":"13.7 Finde den Denkfehler","anker":"sec:dynamische-programmierung-denkfehler","text":"Bei dynamischer Programmierung entscheidet eine einzige Frage über Erfolg oder Misserfolg: Was gehört in den Zustand? Ist er zu klein, rechnet das Verfahren völlig korrekt — nur eb"},{"seite":"dynamische-programmierung.html","titel":"13.8 Micro-Quiz","anker":"sec:dynamische-programmierung-quiz","text":""},{"seite":"dynamische-programmierung.html","titel":"13.9 Selbsttest","anker":"sec:dynamische-programmierung-selbsttest","text":""},{"seite":"dynamische-programmierung.html","titel":"13.10 Zusammenfassung","anker":"sec:dynamische-programmierung-zusammenfassung","text":""},{"seite":"mehrziel.html","titel":"Kapitel 14: Mehrere Ziele — Pareto-Fronten statt Gewichte","anker":null,"text":""},{"seite":"mehrziel.html","titel":"14.1 In 5 Minuten gelöst","anker":"sec:mehrziel-schnellstart","text":""},{"seite":"mehrziel.html","titel":"14.2 Lernziele","anker":"sec:mehrziel-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"mehrziel.html","titel":"14.3 Wann ein Kompromiss gut ist","anker":"sec:mehrziel-dominanz","text":"Bei einem Ziel ist „besser” eindeutig. Bei zwei Zielen nicht mehr: Ein Plan kann billiger und schmutziger sein als ein anderer, und dann ist keiner von beiden besser."},{"seite":"mehrziel.html","titel":"14.4 Der Reflex: alles in ein Ziel rühren","anker":"sec:mehrziel-skalarisierung","text":"Der naheliegende Weg, zwei Ziele zu einem zu machen, ist die gewichtete Summe:"},{"seite":"mehrziel.html","titel":"14.5 Das Programm","anker":"sec:mehrziel-programm","text":"Zwölf Sendungen, drei Verkehrsträger, fünf freie Bahntrassen. Der Zielkonflikt entsteht nicht zwischen den Trägern — die Bahn ist immer billiger und sauberer als der LKW —, sondern"},{"seite":"mehrziel.html","titel":"14.6 Was die gewichtete Summe nicht sieht","anker":"sec:mehrziel-luecke","text":"Die Front hat zehn Punkte. Ein Raster aus 1 401 Gewichten findet sechs davon. Vier Kompromisse sind für kein Gewicht erreichbar — und das ist keine Frage der Rasterweite."},{"seite":"mehrziel.html","titel":"Das Gewicht ist keine Feineinstellung","anker":"das-gewicht-ist-keine-feineinstellung","text":"Die zweite Beobachtung aus derselben Tabelle: 1 401 Gewichte erzeugen sechs verschiedene Pläne. Weite Gewichtsbereiche liefern dasselbe Ergebnis, und dazwischen springt es."},{"seite":"mehrziel.html","titel":"14.7 Das ε-Constraint-Verfahren","anker":"sec:mehrziel-epsilon","text":"Der Ausweg ist unspektakulär: Man macht aus dem zweiten Ziel eine Nebenbedingung."},{"seite":"mehrziel.html","titel":"Wenn es eine klare Rangfolge gibt: lexikografisch","anker":"wenn-es-eine-klare-rangfolge-gibt-lexikografisch","text":"Manchmal ist ein Ziel wirklich vorrangig. Dann optimiert man lexikografisch: erst das erste Ziel, dann das zweite unter der Bedingung, dass das erste (fast) erhalten bleibt."},{"seite":"mehrziel.html","titel":"14.8 Wie man die Front vorlegt","anker":"sec:mehrziel-vorlegen","text":"Eine Front mit zehn Punkten ist kein Ergebnis, sondern eine Speisekarte. Damit sie entscheidbar wird, gehört eine Spalte dazu, die im Programm EUR je kg heißt:"},{"seite":"mehrziel.html","titel":"14.9 Übungsaufgaben","anker":"sec:mehrziel-uebungsaufgaben","text":""},{"seite":"mehrziel.html","titel":"14.10 Finde den Denkfehler","anker":"sec:mehrziel-denkfehler","text":""},{"seite":"mehrziel.html","titel":"14.11 Micro-Quiz","anker":"sec:mehrziel-quiz","text":""},{"seite":"mehrziel.html","titel":"14.12 Selbsttest","anker":"sec:mehrziel-selbsttest","text":""},{"seite":"mehrziel.html","titel":"14.13 Zusammenfassung","anker":"sec:mehrziel-zusammenfassung","text":""},{"seite":"prognose.html","titel":"Kapitel 15: Predict-then-Optimize — die bessere Prognose, die schlechtere Entscheidung","anker":null,"text":""},{"seite":"prognose.html","titel":"15.1 In 5 Minuten gelöst","anker":"sec:prognose-schnellstart","text":""},{"seite":"prognose.html","titel":"15.2 Lernziele","anker":"sec:prognose-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"prognose.html","titel":"15.3 Die Naht zwischen zwei Abteilungen","anker":"sec:prognose-naht","text":"In fast jedem Unternehmen sieht der Ablauf so aus:"},{"seite":"prognose.html","titel":"15.4 Das Programm","anker":"sec:prognose-programm","text":"Dieselbe Bäckerei, aber mit Merkmalen: Wochentag, Temperatur, Aktionstage. Die Nachfrage muss jetzt modelliert werden."},{"seite":"prognose.html","titel":"Was Sie in der Abbildung sehen. Links und rechts gewinnt nicht dasselbe Verfahren. Die Punktprognose hat den besten Prognosefehler und die höchsten Kosten — beide Maßstäbe messen etwas anderes, und nur einer davon steht am Ende in der Bilanz.","anker":"was-sie-in-der-abbildung-sehen.-links-und-rechts-gewinnt-nicht-dasselbe-verfahren.-die-punktprognose-hat-den-besten-prognosefehler-und-die-höchsten-kosten-beide-maßstäbe-messen-etwas-anderes-und-nur-einer-davon-steht-am-ende-in-der-bilanz.","text":""},{"seite":"prognose.html","titel":"15.5 Der Befund","anker":"sec:prognose-befund","text":""},{"seite":"prognose.html","titel":"Warum ein pauschaler Zuschlag zu kurz greift","anker":"warum-ein-pauschaler-zuschlag-zu-kurz-greift","text":"Die zweite Zeile der Tabelle ist die verbreitete Praxis: Punktprognose plus Sicherheitsbestand. Sie hilft (−7,3 %), bleibt aber hinter der Quantilregression zurück. Das Programm re"},{"seite":"prognose.html","titel":"15.6 Eine Messfalle, in die der Autor selbst getappt ist","anker":"sec:prognose-messfalle","text":"Der erste Entwurf dieses Programms bewertete auf 230 Testtagen — ein realistischer Zeitraum, gut sieben Monate. Dort hatte die Quantilregression den besseren MSE, und die ganze Aus"},{"seite":"prognose.html","titel":"15.7 Wie weit das Verfahren reicht","anker":"sec:prognose-ausblick","text":"Die dritte Zeile der Ergebnistabelle — „Zuschlag auf Kosten trainiert” — ist die einfachste denkbare Form einer Idee, die derzeit erforscht wird: das Prognosemodell direkt auf die "},{"seite":"prognose.html","titel":"15.8 Übungsaufgaben","anker":"sec:prognose-uebungsaufgaben","text":""},{"seite":"prognose.html","titel":"15.9 Finde den Denkfehler","anker":"sec:prognose-denkfehler","text":""},{"seite":"prognose.html","titel":"15.10 Micro-Quiz","anker":"sec:prognose-quiz","text":""},{"seite":"prognose.html","titel":"15.11 Selbsttest","anker":"sec:prognose-selbsttest","text":""},{"seite":"prognose.html","titel":"15.12 Zusammenfassung","anker":"sec:prognose-zusammenfassung","text":""},{"seite":"synthese-nichtlinear.html","titel":"Synthese Teil III","anker":null,"text":""},{"seite":"synthese-nichtlinear.html","titel":"Welche Annahme fällt weg?","anker":"welche-annahme-fällt-weg","text":""},{"seite":"synthese-nichtlinear.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-nichtlinear.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-nichtlinear.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"bruecke.html","titel":"Kapitel 16: Die Strukturbrücke — dieselbe Mathematik, zwei Welten","anker":null,"text":""},{"seite":"bruecke.html","titel":"16.1 In 5 Minuten gelöst","anker":"sec:bruecke-schnellstart","text":""},{"seite":"bruecke.html","titel":"16.2 Lernziele","anker":"sec:bruecke-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"bruecke.html","titel":"16.3 Das Programm","anker":"sec:bruecke-programm","text":""},{"seite":"bruecke.html","titel":"16.4 Die zwei Brücken im Einzelnen","anker":"sec:bruecke-einzelnen","text":""},{"seite":"bruecke.html","titel":"Erste Brücke: Allokation","anker":"erste-brücke-allokation","text":""},{"seite":"bruecke.html","titel":"Zweite Brücke: Absicherung gegen den schlechtesten Fall","anker":"zweite-brücke-absicherung-gegen-den-schlechtesten-fall","text":"Hier ist die Übereinstimmung noch enger, weil sie nicht nur die Struktur betrifft, sondern die Formel. Die Rockafellar-Uryasev-Formulierung aus Kapitel 20"},{"seite":"bruecke.html","titel":"Wo die Lösungen sich unterscheiden","anker":"wo-die-lösungen-sich-unterscheiden","text":"Der Vergleich fördert nebenbei einen Unterschied zutage, den man nicht erwartet: Die Streuungsgrenze von 40 % bindet nur im Einkauf, nicht im Depot."},{"seite":"bruecke.html","titel":"16.5 Wo die Brücke endet","anker":"sec:bruecke-grenzen","text":"Eine Analogie ist nur so nützlich wie die Grenzen, die man ihr zieht. Drei Dinge wandern nicht mit:"},{"seite":"bruecke.html","titel":"16.6 Übungsaufgaben","anker":"sec:bruecke-uebungsaufgaben","text":""},{"seite":"bruecke.html","titel":"16.7 Finde den Denkfehler","anker":"sec:bruecke-denkfehler","text":""},{"seite":"bruecke.html","titel":"16.8 Micro-Quiz","anker":"sec:bruecke-quiz","text":""},{"seite":"bruecke.html","titel":"16.9 Selbsttest","anker":"sec:bruecke-selbsttest","text":""},{"seite":"bruecke.html","titel":"16.10 Zusammenfassung","anker":"sec:bruecke-zusammenfassung","text":""},{"seite":"supplychain.html","titel":"Kapitel 17: Supply-Chain und Energieeinsatz unter Unsicherheit","anker":null,"text":""},{"seite":"supplychain.html","titel":"17.1 In 5 Minuten gelöst","anker":"sec:supplychain-schnellstart","text":""},{"seite":"supplychain.html","titel":"17.2 Lernziele","anker":"sec:supplychain-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"supplychain.html","titel":"17.3 Die Aufgabe","anker":"sec:supplychain-aufgabe","text":"Fünf Blöcke, 24 Stunden, eine Last zwischen 270 und 804 MW. Dazu Windeinspeisung, die niemand am Vorabend kennt."},{"seite":"supplychain.html","titel":"Die Zeitstruktur der Entscheidung","anker":"die-zeitstruktur-der-entscheidung","text":"Hier geht das Kapitel über Abschnitt 12.5 hinaus:"},{"seite":"supplychain.html","titel":"17.4 Das Programm","anker":"sec:supplychain-programm","text":""},{"seite":"supplychain.html","titel":"17.5 Der Befund","anker":"sec:supplychain-befund","text":""},{"seite":"supplychain.html","titel":"17.6 Was Versorgungssicherheit kostet","anker":"sec:supplychain-sicherheit","text":"Der dritte Plan legt eine CVaR-Schranke auf die nicht gedeckte Energie: Im Mittel der schlechtesten 10 % der Szenarien darf nichts fehlen. Dieselbe Rockafellar-Uryasev-Konstruktion"},{"seite":"supplychain.html","titel":"17.7 Übungsaufgaben","anker":"sec:supplychain-uebungsaufgaben","text":""},{"seite":"supplychain.html","titel":"17.8 Finde den Denkfehler","anker":"sec:supplychain-denkfehler","text":""},{"seite":"supplychain.html","titel":"17.9 Micro-Quiz","anker":"sec:supplychain-quiz","text":""},{"seite":"supplychain.html","titel":"17.10 Selbsttest","anker":"sec:supplychain-selbsttest","text":""},{"seite":"supplychain.html","titel":"17.11 Zusammenfassung","anker":"sec:supplychain-zusammenfassung","text":""},{"seite":"finanzdaten.html","titel":"Kapitel 18: Finanzdaten-Modellierung — Renditen, Kovarianz und Shrinkage","anker":null,"text":""},{"seite":"finanzdaten.html","titel":"18.1 In 5 Minuten gelöst","anker":"sec:finanzdaten-schnellstart","text":""},{"seite":"finanzdaten.html","titel":"18.2 Lernziele","anker":"sec:finanzdaten-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"finanzdaten.html","titel":"18.3 Diskrete und logarithmische Renditen","anker":"sec:finanzdaten-diskrete-und-logarithmische-renditen","text":"Sei P_{i,t} der bereinigte Schlusskurs (adjusted close) von Titel i zum Zeitpunkt t. „Bereinigt“ heißt: um Dividenden und Aktiensplits korrigiert — sonst erscheint jede Dividendenz"},{"seite":"finanzdaten.html","titel":"Diskrete Rendite","anker":"diskrete-rendite","text":"R_{i,t} = \\frac{P_{i,t} - P_{i,t-1}}{P_{i,t-1}} = \\frac{P_{i,t}}{P_{i,t-1}} - 1"},{"seite":"finanzdaten.html","titel":"Logarithmische Rendite","anker":"logarithmische-rendite","text":"r_{i,t} = \\ln\\!\\left(\\frac{P_{i,t}}{P_{i,t-1}}\\right) = \\ln P_{i,t} - \\ln P_{i,t-1}"},{"seite":"finanzdaten.html","titel":"18.4 Das Schätzfehler-Problem","anker":"sec:finanzdaten-das-schaetzfehler-problem","text":"Sei \\mathbf{X}\\in\\mathbb{R}^{T\\times N} die Matrix zentrierter Renditen von N Titeln über T Handelstage. Die Stichproben-Kovarianzmatrix lautet:"},{"seite":"finanzdaten.html","titel":"18.5 Ledoit-Wolf-Shrinkage","anker":"sec:finanzdaten-ledoit-wolf-shrinkage","text":"Olivier Ledoit und Michael Wolf lösten das Dilemma durch lineare Schrumpfung:"},{"seite":"finanzdaten.html","titel":"Welches Ziel \\mathbf{F}?","anker":"welches-ziel-mathbff","text":"Ein häufiges Missverständnis: Es gibt nämlich mehrere Ledoit-Wolf-Varianten, und Text und Code müssen sich konsistent auf dieselbe beziehen:"},{"seite":"finanzdaten.html","titel":"18.6 Praxis: Datenpipeline mit korrekter Spaltenreihenfolge","anker":"sec:finanzdaten-praxis-datenpipeline-mit-korrekter-spaltenreihenfolge","text":""},{"seite":"finanzdaten.html","titel":"18.7 Übungsaufgaben","anker":"sec:finanzdaten-uebungsaufgaben","text":""},{"seite":"finanzdaten.html","titel":"18.8 Finde den Denkfehler","anker":"sec:finanzdaten-denkfehler","text":""},{"seite":"finanzdaten.html","titel":"18.9 Micro-Quiz","anker":"sec:finanzdaten-quiz","text":""},{"seite":"finanzdaten.html","titel":"18.10 Selbsttest","anker":"sec:finanzdaten-selbsttest","text":""},{"seite":"finanzdaten.html","titel":"18.11 Zusammenfassung","anker":"sec:finanzdaten-zusammenfassung","text":""},{"seite":"markowitz.html","titel":"Kapitel 19: Die moderne Portfoliotheorie nach Markowitz","anker":null,"text":""},{"seite":"markowitz.html","titel":"19.1 In 5 Minuten gelöst","anker":"sec:markowitz-schnellstart","text":""},{"seite":"markowitz.html","titel":"19.2 Lernziele","anker":"sec:markowitz-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"markowitz.html","titel":"19.3 Warum Diversifikation funktioniert","anker":"sec:markowitz-warum-diversifikation-funktioniert","text":"Harry Markowitz’ zentrale Einsicht von 1952: Das Risiko eines Portfolios ist nicht die Summe der Einzelrisiken. Es hängt davon ab, wie die Titel zusammenspielen."},{"seite":"markowitz.html","titel":"19.4 Das Mean-Variance-Modell","anker":"sec:markowitz-das-mean-variance-modell","text":"Sei \\mathbf{w} = (w_1,\\dots,w_n)^\\top der Vektor der Portfoliogewichte:"},{"seite":"markowitz.html","titel":"Die Effizienzgrenze und das Tangentialportfolio","anker":"die-effizienzgrenze-und-das-tangentialportfolio","text":"Die Effizienzgrenze (efficient frontier) ist die Kurve aller Portfolios, die zu jedem Risikoniveau die höchstmögliche Rendite liefern."},{"seite":"markowitz.html","titel":"Die Korn-Transformation","anker":"die-korn-transformation","text":"Die Sharpe Ratio ist ein Bruch — und damit nicht konvex. CVXPY würde sie ablehnen. Der Ausweg ist ein eleganter Trick, der die Homogenität des Problems ausnutzt:"},{"seite":"markowitz.html","titel":"19.5 Institutionelle Nebenbedingungen","anker":"sec:markowitz-institutionelle-nebenbedingungen","text":"In der Praxis gelten regulatorische und interne Regeln:"},{"seite":"markowitz.html","titel":"Was diese Regeln kosten","anker":"was-diese-regeln-kosten","text":"Jede Regel schneidet etwas vom Erreichbaren ab. Wie viel, lässt sich ausrechnen — man legt die Effizienzlinie ohne und mit Beschränkungen übereinander:"},{"seite":"markowitz.html","titel":"19.6 Vollimplementierung mit CVXPY","anker":"sec:markowitz-vollimplementierung-mit-cvxpy","text":""},{"seite":"markowitz.html","titel":"19.7 Übungsaufgaben","anker":"sec:markowitz-uebungsaufgaben","text":""},{"seite":"markowitz.html","titel":"19.8 Finde den Denkfehler","anker":"sec:markowitz-denkfehler","text":"Kapitel 18 hat gezeigt, was eine schlecht geschätzte Kovarianzmatrix anrichtet. Bei den erwarteten Renditen ist es schlimmer — und weil sie in der Zielfunktion stehen statt in der "},{"seite":"markowitz.html","titel":"19.9 Micro-Quiz","anker":"sec:markowitz-quiz","text":""},{"seite":"markowitz.html","titel":"19.10 Selbsttest","anker":"sec:markowitz-selbsttest","text":""},{"seite":"markowitz.html","titel":"19.11 Zusammenfassung","anker":"sec:markowitz-zusammenfassung","text":""},{"seite":"cvar.html","titel":"Kapitel 20: Tail-Risiko, CVaR und Transaktionskosten","anker":null,"text":""},{"seite":"cvar.html","titel":"20.1 In 5 Minuten gelöst","anker":"sec:cvar-schnellstart","text":""},{"seite":"cvar.html","titel":"20.2 Lernziele","anker":"sec:cvar-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"cvar.html","titel":"20.3 Die zwei Schwächen des Markowitz-Modells","anker":"sec:cvar-die-zwei-schwaechen-des-markowitz-modells","text":"Erstens: die Normalverteilungs-Illusion. Die Varianz behandelt Aufwärts- und Abwärtsschwankungen gleich und unterstellt implizit symmetrische, dünn auslaufende Verteilungen. Reale "},{"seite":"cvar.html","titel":"20.4 Value at Risk und Conditional Value at Risk","anker":"sec:cvar-value-at-risk-und-conditional-value","text":""},{"seite":"cvar.html","titel":"Das Rockafellar-Uryasev-Theorem (2000)","anker":"das-rockafellar-uryasev-theorem-2000","text":"Rockafellar und Uryasev zeigten, dass sich der CVaR über S Szenarien exakt als lineares Programm formulieren lässt:"},{"seite":"cvar.html","titel":"20.5 Transaktionskosten über die L_1-Norm","anker":"sec:cvar-transaktionskosten-ueber-die-l-1-norm","text":"Sei \\mathbf{w}_{\\text{alt}} das bestehende Portfolio und \\mathbf{w} das neue Ziel. Der Umschlag (turnover) ist:"},{"seite":"cvar.html","titel":"20.6 Implementierung: CVaR-Portfolio mit Reibung","anker":"sec:cvar-implementierung-cvar-portfolio-mit-reibung","text":""},{"seite":"cvar.html","titel":"20.7 Übungsaufgaben","anker":"sec:cvar-uebungsaufgaben","text":""},{"seite":"cvar.html","titel":"20.8 Finde den Denkfehler","anker":"sec:cvar-denkfehler","text":""},{"seite":"cvar.html","titel":"20.9 Micro-Quiz","anker":"sec:cvar-quiz","text":""},{"seite":"cvar.html","titel":"20.10 Selbsttest","anker":"sec:cvar-selbsttest","text":""},{"seite":"cvar.html","titel":"20.11 Zusammenfassung","anker":"sec:cvar-zusammenfassung","text":""},{"seite":"handelsmaschine.html","titel":"Kapitel 21: Die vollständige quantitative Handelsmaschine","anker":null,"text":""},{"seite":"handelsmaschine.html","titel":"21.1 In 5 Minuten gelöst","anker":"sec:handelsmaschine-schnellstart","text":""},{"seite":"handelsmaschine.html","titel":"21.2 Lernziele","anker":"sec:handelsmaschine-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"handelsmaschine.html","titel":"21.3 Die Architektur","anker":"sec:handelsmaschine-die-architektur","text":""},{"seite":"handelsmaschine.html","titel":"21.4 Rebalancing-Termine richtig bestimmen","anker":"sec:handelsmaschine-rebalancing-termine-richtig-bestimmen","text":""},{"seite":"handelsmaschine.html","titel":"21.5 Die Engine","anker":"sec:handelsmaschine-die-engine","text":""},{"seite":"handelsmaschine.html","titel":"21.6 Die fünf Selbsttäuschungen des Backtestens","anker":"sec:handelsmaschine-die-fuenf-selbsttaeuschungen-des-backtestens","text":""},{"seite":"handelsmaschine.html","titel":"21.7 Vom Backtest zum Betrieb","anker":"sec:handelsmaschine-vom-backtest-zum-betrieb","text":"Ein guter Backtest ist eine notwendige, keine hinreichende Bedingung. Was zwischen Backtest und echtem Einsatz liegt:"},{"seite":"handelsmaschine.html","titel":"21.8 Übungsaufgaben","anker":"sec:handelsmaschine-uebungsaufgaben","text":""},{"seite":"handelsmaschine.html","titel":"21.9 Finde den Denkfehler","anker":"sec:handelsmaschine-denkfehler","text":"Backtest_Fallen.py zeigt drei Fehler, die im Backtest stecken: Lookahead, Survivorship, vergessene Kosten. Alle drei kann man einem Programm ansehen, wenn man es liest."},{"seite":"handelsmaschine.html","titel":"21.10 Micro-Quiz","anker":"sec:handelsmaschine-quiz","text":""},{"seite":"handelsmaschine.html","titel":"21.11 Selbsttest","anker":"sec:handelsmaschine-selbsttest","text":""},{"seite":"handelsmaschine.html","titel":"21.12 Zusammenfassung","anker":"sec:handelsmaschine-zusammenfassung","text":""},{"seite":"synthese-anwendungen.html","titel":"Synthese Teil IV","anker":null,"text":""},{"seite":"synthese-anwendungen.html","titel":"Dieselbe Struktur, andere Namen","anker":"dieselbe-struktur-andere-namen","text":""},{"seite":"synthese-anwendungen.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-anwendungen.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-anwendungen.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"praxisfallen.html","titel":"Kapitel 22: Praxisfallen und der Weg zum produktiven Einsatz","anker":null,"text":""},{"seite":"praxisfallen.html","titel":"22.1 In 5 Minuten gelöst","anker":"sec:praxisfallen-schnellstart","text":"Die vorangegangenen vierzehn Kapitel enden fast alle mit derselben Empfehlung: Prüfen Sie das Ergebnis gegen die Wirklichkeit. Dieses Kapitel beginnt damit, diese Prüfung als wiede"},{"seite":"praxisfallen.html","titel":"22.2 Lernziele","anker":"sec:praxisfallen-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"praxisfallen.html","titel":"22.3 Die fünf typischen Praxisfallen","anker":"sec:praxisfallen-die-fuenf-typischen-praxisfallen","text":""},{"seite":"praxisfallen.html","titel":"Falle 1 — Infeasibility","anker":"falle-1-infeasibility","text":"Wenn Parameter unglücklich zusammentreffen — drei Mitarbeitende gleichzeitig krank, kein qualifizierter Ersatz —, meldet der Solver schlicht INFEASIBLE."},{"seite":"praxisfallen.html","titel":"Falle 2 — Der Black-Box-Effekt","anker":"falle-2-der-black-box-effekt","text":"Ein Planer oder Trader lehnt ein System ab, wenn er die Zuweisungen nicht nachvollziehen kann. Akzeptanz ist keine Nebensache, sondern Voraussetzung für den Betrieb."},{"seite":"praxisfallen.html","titel":"Falle 3 — Regimewechsel und Schätzfehler","anker":"falle-3-regimewechsel-und-schätzfehler","text":"Ein Modell, das auf einer mehrjährigen Aufwärtsphase kalibriert wurde, kennt keine Liquiditätskrise und keine Zinswende."},{"seite":"praxisfallen.html","titel":"Falle 4 — Lookahead- und Survivorship-Bias","anker":"falle-4-lookahead--und-survivorship-bias","text":"Ausführlich in Kapitel 21 behandelt. Die Kurzfassung: Beide Fehler machen Backtests systematisch zu gut, ohne eine Fehlermeldung zu erzeugen. Der Lookahead-Selbsttest aus Abschnitt"},{"seite":"praxisfallen.html","titel":"Falle 5 — Laufzeitexplosion","anker":"falle-5-laufzeitexplosion","text":"Exakte Optimalität kann bei MILP oder CP-SAT Minuten bis Stunden dauern — und die Laufzeit wächst nicht linear mit der Problemgröße."},{"seite":"praxisfallen.html","titel":"22.4 Constraint Attribution: welche Bedingung kostet wie viel?","anker":"sec:praxisfallen-attribution","text":"Der Constraint-Trace beantwortet „warum diese Zuweisung?“. Der Deletion Filter in Anhang C beantwortet „warum geht es gar nicht?“. Der häufigste Fall in der Praxis liegt dazwischen"},{"seite":"praxisfallen.html","titel":"Drei Befunde, und keiner steht im Solverergebnis","anker":"drei-befunde-und-keiner-steht-im-solverergebnis","text":"Erstens: bindend heißt nicht teuer. Fünf Bedingungen binden, aber „Lackierbudget Schicht 2“ tut es mit einem Schattenpreis von 0,00 €. Sie berührt den Plan, ohne ihn zu verteuern —"},{"seite":"praxisfallen.html","titel":"Vom Zahlenwerk zur Vorlage","anker":"vom-zahlenwerk-zur-vorlage","text":"Alle drei Antworten stehen bis hierhin in Tabellen. Tabellen liest niemand in einer Abteilungsleiterrunde. Teil (4) des Programms baut aus denselben Zahlen einen Text — und das ist"},{"seite":"praxisfallen.html","titel":"22.5 Architektur einer produktionsreifen OR-Plattform","anker":"sec:praxisfallen-architektur-einer-produktionsreifen-or-plattform","text":""},{"seite":"praxisfallen.html","titel":"22.6 Der gemeinsame Unterbau: or_kern.py","anker":"sec:praxisfallen-or-kern","text":"Die fünf Prinzipien oben beschreiben eine Plattform. Der Weg dorthin beginnt aber viel kleiner — mit der Frage, welche Teile eines Optimierungsprogramms immer dieselben sind."},{"seite":"praxisfallen.html","titel":"Die vier Schichten","anker":"die-vier-schichten","text":"\\underbrace{\\text{Rohdaten}}_{\\text{Excel, CSV, ERP}}\n\\;\\longrightarrow\\;\n\\underbrace{\\text{Domänenmodell}}_{\\text{geprüft, solverfrei}}\n\\;\\longrightarrow\\;\n\\underbrace{\\text{Model"},{"seite":"praxisfallen.html","titel":"Ein Enum für fünf Bibliotheken","anker":"ein-enum-für-fünf-bibliotheken","text":"Der unscheinbarste, aber wirksamste Teil ist die Statusübersetzung. Dieselbe Aussage heißt in den fünf im Buch verwendeten Bibliotheken:"},{"seite":"praxisfallen.html","titel":"22.7 Der Solverwechsel in der Praxis","anker":"sec:praxisfallen-solverwechsel","text":"Der vorige Abschnitt endet mit einer Behauptung: Ein Solverwechsel betreffe genau einen Baustein. Das ist die Art Satz, die in jedem Architekturvortrag vorkommt und die man deshalb"},{"seite":"praxisfallen.html","titel":"22.8 Eine Checkliste vor dem Produktivgang","anker":"sec:praxisfallen-eine-checkliste-vor-dem-produktivgang","text":"Arbeiten Sie diese Liste ab, bevor ein OR-System in Betrieb geht:"},{"seite":"praxisfallen.html","titel":"22.9 Weiterführende Literatur und Roadmap","anker":"sec:praxisfallen-weiterfuehrende-literatur-und-roadmap","text":"Grundlagen und lineare/ganzzahlige Optimierung * Bertsimas &amp; Tsitsiklis: Introduction to Linear Optimization — Standardwerk zu Simplex, Dualität, Polyedertheorie. * Wolsey: Int"},{"seite":"praxisfallen.html","titel":"22.10 Übungsaufgaben","anker":"sec:praxisfallen-uebungsaufgaben","text":""},{"seite":"praxisfallen.html","titel":"22.11 Finde den Denkfehler","anker":"sec:praxisfallen-denkfehler","text":"Falle 5 — die Laufzeitexplosion — zeigt sich im Betrieb selten als Absturz. Sie zeigt sich als gar nichts."},{"seite":"praxisfallen.html","titel":"22.12 Micro-Quiz","anker":"sec:praxisfallen-quiz","text":""},{"seite":"praxisfallen.html","titel":"22.13 Selbsttest","anker":"sec:praxisfallen-selbsttest","text":""},{"seite":"praxisfallen.html","titel":"22.14 Zusammenfassung","anker":"sec:praxisfallen-zusammenfassung","text":""},{"seite":"testing.html","titel":"Kapitel 23: Testen, Messen, Ausliefern","anker":null,"text":""},{"seite":"testing.html","titel":"23.1 In 5 Minuten gelöst","anker":"sec:testing-schnellstart","text":""},{"seite":"testing.html","titel":"23.2 Lernziele","anker":"sec:testing-lernziele","text":"Nach diesem Kapitel können Sie …"},{"seite":"testing.html","titel":"23.3 Warum Optimierungsmodelle schwer zu testen sind","anker":"sec:testing-warum-schwer","text":"Bei einer gewöhnlichen Funktion schreibt man den erwarteten Wert hin:"},{"seite":"testing.html","titel":"23.4 Die Testsuite","anker":"sec:testing-suite","text":""},{"seite":"testing.html","titel":"23.5 Wer testet die Tests?","anker":"sec:testing-mutation","text":"33 grüne Tests. Das ist noch kein Ergebnis."},{"seite":"testing.html","titel":"Der erste Lauf fand zwei Lücken","anker":"der-erste-lauf-fand-zwei-lücken","text":"Die Suite hatte zunächst 31 Tests. Der Mutationstest sagte dazu:"},{"seite":"testing.html","titel":"23.6 Ein Vergleich, dem man glauben kann","anker":"sec:testing-benchmark","text":"Solververgleiche stehen in jedem Blog, und die meisten sind wertlos. Nicht, weil falsch gemessen wurde, sondern weil zu wenig dazugesagt wird."},{"seite":"testing.html","titel":"Die interessanteste Spalte heißt „Anteil”","anker":"die-interessanteste-spalte-heißt-anteil","text":"Sie sagt, wie viel der Zeit in den Aufbau des Modells geht statt ins Lösen — und die Zahlen sind unbequem:"},{"seite":"testing.html","titel":"23.7 Das Modell als Dienst","anker":"sec:testing-dienst","text":"Ein Modell hinter eine HTTP-Schnittstelle zu hängen sieht nach einer Fingerübung aus. Es gibt aber einen Unterschied, der die ganze Bauform bestimmt:"},{"seite":"testing.html","titel":"Threads oder Prozesse? Eine Messung, keine Meinung","anker":"threads-oder-prozesse-eine-messung-keine-meinung","text":"Punkt 6 des Selbsttests misst, ob vier Solverläufe in Threads sich überlappen. Das Ergebnis ist ein Faktor von rund 3,5 bei vier Threads: OR-Tools rechnet in C++ und gibt den GIL w"},{"seite":"testing.html","titel":"Was im echten Betrieb dazukommt","anker":"was-im-echten-betrieb-dazukommt","text":"Das Beispiel hält das Auftragsbuch in einem dict und rechnet in Threads desselben Prozesses. Für einen einzelnen Rechner reicht das; darüber hinaus nicht:"},{"seite":"testing.html","titel":"23.8 Übungsaufgaben","anker":"sec:testing-uebungsaufgaben","text":""},{"seite":"testing.html","titel":"23.9 Finde den Denkfehler","anker":"sec:testing-denkfehler","text":""},{"seite":"testing.html","titel":"23.10 Micro-Quiz","anker":"sec:testing-quiz","text":""},{"seite":"testing.html","titel":"23.11 Selbsttest","anker":"sec:testing-selbsttest","text":""},{"seite":"testing.html","titel":"23.12 Zusammenfassung","anker":"sec:testing-zusammenfassung","text":""},{"seite":"synthese-praxis.html","titel":"Synthese Teil V","anker":null,"text":""},{"seite":"synthese-praxis.html","titel":"Was schiefgeht, und was dagegen hilft","anker":"was-schiefgeht-und-was-dagegen-hilft","text":""},{"seite":"synthese-praxis.html","titel":"Was dieser Teil gemessen hat","anker":"was-dieser-teil-gemessen-hat","text":""},{"seite":"synthese-praxis.html","titel":"Drei Fehler, die dieser Teil verhindert","anker":"drei-fehler-die-dieser-teil-verhindert","text":""},{"seite":"synthese-praxis.html","titel":"Wenn Sie nur eines mitnehmen","anker":"wenn-sie-nur-eines-mitnehmen","text":""},{"seite":"projektwerkstatt.html","titel":"Projektwerkstatt","anker":null,"text":""},{"seite":"projektwerkstatt.html","titel":"Wie Sie ein Projekt bearbeiten","anker":"wie-sie-ein-projekt-bearbeiten","text":"Halten Sie sich an diese Reihenfolge — sie ist aus vielen gescheiterten und einigen gelungenen Projekten destilliert:"},{"seite":"projektwerkstatt.html","titel":"P1 — Vertretungsplaner für eine Schule","anker":"p1-vertretungsplaner-für-eine-schule","text":""},{"seite":"projektwerkstatt.html","titel":"P2 — Schichtplanung für ein Pflegeteam","anker":"p2-schichtplanung-für-ein-pflegeteam","text":""},{"seite":"projektwerkstatt.html","titel":"P3 — Tourenplanung für einen Lieferdienst","anker":"p3-tourenplanung-für-einen-lieferdienst","text":""},{"seite":"projektwerkstatt.html","titel":"P4 — Standort- und Lagernetzplanung","anker":"p4-standort--und-lagernetzplanung","text":""},{"seite":"projektwerkstatt.html","titel":"P5 — Produktionsplanung mit Rüstkosten","anker":"p5-produktionsplanung-mit-rüstkosten","text":""},{"seite":"projektwerkstatt.html","titel":"P6 — Portfolio-Rebalancer für ein Privatdepot","anker":"p6-portfolio-rebalancer-für-ein-privatdepot","text":""},{"seite":"projektwerkstatt.html","titel":"P7 — Risikoreport mit CVaR und Stresstests","anker":"p7-risikoreport-mit-cvar-und-stresstests","text":""},{"seite":"projektwerkstatt.html","titel":"P9 — Wenn der Solver aussteigt: Tourenplanung in Echtgröße","anker":"p9-wenn-der-solver-aussteigt-tourenplanung-in-echtgröße","text":""},{"seite":"projektwerkstatt.html","titel":"P10 — Zwei Ziele, eine Entscheidung: Kosten gegen CO₂","anker":"p10-zwei-ziele-eine-entscheidung-kosten-gegen-co₂","text":""},{"seite":"projektwerkstatt.html","titel":"P11 — Vom Skript zum Dienst: das Modell übergeben","anker":"p11-vom-skript-zum-dienst-das-modell-übergeben","text":""},{"seite":"projektwerkstatt.html","titel":"P8 — Freies Projekt aus Ihrem eigenen Umfeld","anker":"p8-freies-projekt-aus-ihrem-eigenen-umfeld","text":""},{"seite":"projektwerkstatt.html","titel":"Eine Bitte zum Schluss","anker":"eine-bitte-zum-schluss","text":"Wenn Sie ein Projekt fertig haben, machen Sie zwei Dinge:"},{"seite":"anhang-loesungen.html","titel":"Anhang A: Lösungen zu allen Übungsaufgaben","anker":null,"text":""},{"seite":"anhang-loesungen.html","titel":"A.1 Lösungen zu Kapitel „Einführung in Operations Research — Vom Ursprung zur mathematischen Entscheidungsfindung“","anker":"sec:loesungen-einfuehrung","text":"1.1 — Analytik-Stufen. (a) deskriptiv (Vergangenheit beschreiben). (b) prädiktiv (Prognose). (c) präskriptiv (Handlungsanweisung unter Restriktionen). (d) prädiktiv (Klassifikation"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Schreinerei verdoppelt ihren Gewinn","anker":"finde-den-denkfehler-die-schreinerei-verdoppelt-ihren-gewinn","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-einfuehrung","text":"1 — (c) präskriptiv. Die Frage lautet „was tun“, und es gibt eine Restriktion (das Restbudget). Dass man für die Bewertung eine Prognose braucht, macht die Frage nicht prädiktiv — "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-einfuehrung","text":""},{"seite":"anhang-loesungen.html","titel":"A.2 Lösungen zu Kapitel „Das mathematische Fundament — Vektoren, Matrizen, Konvexität“","anker":"sec:loesungen-fundament","text":"2.1 — Matrixform lesen. \\max 4x_1 + x_2 + 6x_3 u. d. N. x_1 + 2x_2 \\le 10, x_2 + 3x_3 \\le 12, x \\ge 0. 3 Variablen, 2 Nebenbedingungen (plus Nichtnegativität)."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die unauffällige Transposition","anker":"finde-den-denkfehler-die-unauffällige-transposition","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-fundament","text":"1 — (b). Der Fundamentalsatz sagt nur, dass ein Optimum in einer Ecke angenommen wird. (a) ist falsch, weil es mehrere optimale Ecken geben kann (wenn die Zielfunktion parallel zu "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-fundament","text":""},{"seite":"anhang-loesungen.html","titel":"A.3 Lösungen zu Kapitel „Das Python-Ökosystem für OR — Solver, Bindings und Modellierungsschichten“","anker":"sec:loesungen-oekosystem","text":"3.1 — Solverwahl. (a) CP-SAT (diskrete Zuordnung mit Zeitfenstern). (b) scipy.optimize.linprog (klassisches Mischungs-LP, klein). (c) CVXPY (konvexes QP). (d) MIQP — CVXPY mit Binä"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Der Solver, der angeblich dreimal schneller ist","anker":"finde-den-denkfehler-der-solver-der-angeblich-dreimal-schneller-ist","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-oekosystem","text":"1 — (b) CP-SAT. Es geht um diskrete Zuweisung mit logischen Regeln („nicht gleichzeitig“) — CP-SATs Kerngebiet, und Regeln dieser Art formuliert man dort direkt statt über Big-M-Um"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-oekosystem","text":""},{"seite":"anhang-loesungen.html","titel":"A.4 Lösungen zu Kapitel „Vom Management-Wunsch zum Modell“","anker":"sec:loesungen-modellierung","text":"4.1 — Die fehlende Lesart. Eine dritte Lesart: „Der Umsatzanteil der Stammkunden darf nicht unter 30 % fallen.” Sie zielt weder auf einzelne Aufträge noch auf einzelne Kunden, sond"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Die Engpassmaschine soll zu mindestens 92 % ausgelastet sein“","anker":"denkfehler-loesung-modellierung","text":"Fehler 1: Auslastung ist ein Ergebnis, keine Anforderung."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-modellierung","text":"1. b) Die Zielfunktion ist eine vollständige Aussage darüber, was zählt. Was nicht darin steht, ist für das Modell nicht „weniger wichtig”, sondern wertlos. (a) und (c) sind frei e"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-modellierung","text":""},{"seite":"anhang-loesungen.html","titel":"A.5 Lösungen zu Kapitel „Lineare Programmierung — Simplex, Dualität und Schattenpreise“","anker":"sec:loesungen-lp","text":"5.1 — Schlupf deuten. (a) Ressourcen 1 und 3 (Schlupf 0). (b) Ja: Überall ist s_i \\cdot y_i = 0. (c) In Ressource 3 — der höchste Schattenpreis (9,8) bedeutet den größten Grenznutz"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die 380 000-Euro-Maschine","anker":"finde-den-denkfehler-die-380-000-euro-maschine","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-lp","text":"1 — (b). Genau der Satz vom komplementären Schlupf: s_i \\cdot y_i = 0. Ist Schlupf vorhanden, muss der Dualwert null sein. (a) hat die Aussage in ihr Gegenteil verkehrt; (c) verwec"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-lp","text":""},{"seite":"anhang-loesungen.html","titel":"A.6 Lösungen zu Kapitel „Gemischt-ganzzahlige Optimierung — Diskrete Entscheidungen und Branch-and-Bound“","anker":"sec:loesungen-milp","text":"6.1 — Runden widerlegen. Beispiel: \\max x_1 + x_2 u. d. N. 10x_1 + 10x_2 \\le 15, ganzzahlig. LP-Optimum Z = 1{,}5; Abrunden ergibt (0,0) mit Z = 0 — 100 % Verlust. Das ganzzahlige "},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Elf Lager, die keine Fixkosten kosten","anker":"finde-den-denkfehler-elf-lager-die-keine-fixkosten-kosten","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-milp","text":"1 — (b). Der Gap (48\\,200 - 47\\,100)/48\\,200 = 2{,}3\\,\\% ist eine Garantie, keine Schätzung: Besser als 47 100 € kann es nachweislich nicht werden, also ist der vorliegende Plan hö"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-milp","text":""},{"seite":"anhang-loesungen.html","titel":"A.7 Lösungen zu Kapitel „Constraint Programming mit CP-SAT — Logik, Scheduling und Zuweisung“","anker":"sec:loesungen-cpsat","text":"7.1 — Propagation. Aus x_1 + x_2 = 8 und x_1 &lt; x_2 folgt x_1 &lt; 4, also x_1 \\in \\{2,3\\} (denn x_2 = 8-x_1 \\le 6 verlangt x_1 \\ge 2) und entsprechend x_2 \\in \\{5,6\\}. Aus 36 Ko"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Betriebsvereinbarung, die niemanden interessiert","anker":"finde-den-denkfehler-die-betriebsvereinbarung-die-niemanden-interessiert","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-cpsat","text":"1 — (b) wirkungslos. INFEASIBLE ist kein Abbruch, sondern ein Beweis: CP-SAT hat gezeigt, dass keine zulässige Lösung existiert. Mehr Zeit (a) oder mehr Arbeiter (c) ändern daran n"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-cpsat","text":""},{"seite":"anhang-loesungen.html","titel":"A.8 Lösungen zu Kapitel „Graphen, Flüsse und Touren — Min-Cost-Flow, Matching und VRP“","anker":"sec:loesungen-graphen","text":"8.1 — Flusserhaltung. b_i = (15+3) - (12+8) = -2 → Senke (Nettobedarf 2)."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die vergessene Dimension","anker":"finde-den-denkfehler-die-vergessene-dimension","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-graphen","text":"1 — (b) totale Unimodularität. Jede quadratische Teilmatrix hat Determinante 0, +1 oder -1; bei ganzzahliger rechter Seite sind deshalb alle Ecken des zulässigen Bereichs ganzzahli"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-graphen","text":""},{"seite":"anhang-loesungen.html","titel":"A.9 Lösungen zu Kapitel „Metaheuristiken — wenn der exakte Solver aussteigt“","anker":"sec:loesungen-metaheuristiken","text":"9.1 — Die Kurzsichtigkeit von Hand. Ab Schwarz wählt die Faustregel: Schwarz → Dunkelrot (26) → Rot (5) → … und muss dann in die helle Gruppe: Rot → Weiß (50) → Elfenbein (2) → Bei"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Unsere Heuristik ist 6 % besser“","anker":"denkfehler-loesung-metaheuristiken","text":"Es fehlt die Schranke."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-metaheuristiken","text":"1. b) Eine lokale Suche lebt von der Zahl geprüfter Züge. Die volle Summe kostet O(n) statt O(1) — bei 500 Aufträgen zwei Größenordnungen weniger Züge im selben Zeitbudget. Numeris"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-metaheuristiken","text":""},{"seite":"anhang-loesungen.html","titel":"A.10 Lösungen zu Kapitel „Spaltengenerierung“","anker":"sec:loesungen-dekomposition","text":"10.1 — Das Abbruchkriterium. Die Eins ist der Zielfunktionskoeffizient eines Musters: Jedes geschnittene Muster verbraucht genau eine Mutterrolle, und die Zielfunktion lautet \\min "},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Die LP-Lösung sagt 72,92 — also runden wir auf“","anker":"denkfehler-loesung-dekomposition","text":"Das Aufrunden kostet sechs Rollen — acht Prozent."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-dekomposition","text":"1. b) Das Mustermodell kennt keine einzelnen Rollen und damit keine austauschbaren Objekte — die Symmetrie verschwindet. (a) trifft zufällig auch zu, ist aber nicht der Grund; (c) "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-dekomposition","text":""},{"seite":"anhang-loesungen.html","titel":"A.11 Lösungen zu Kapitel „Quadratische und nichtlineare Optimierung — KKT, Lagrange, Konvexität“","anker":"sec:loesungen-qp-nlp","text":"11.1 — Konvexität einordnen. PSD (alle \\ge 0) → konvex, aber nicht streng konvex; die Lösung ist nicht notwendig eindeutig. Der Eigenwert 0 bedeutet eine flache Richtung: Entlang d"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Kovarianzmatrix aus dem Controlling","anker":"finde-den-denkfehler-die-kovarianzmatrix-aus-dem-controlling","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-qp-nlp","text":"1 — (b). Der DCPError ist eine inhaltliche Aussage: Für diese Formulierung kann es keine Optimalitätsgarantie geben. Auf scipy auszuweichen (a) beseitigt die Meldung, nicht die Urs"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-qp-nlp","text":""},{"seite":"anhang-loesungen.html","titel":"A.12 Lösungen zu Kapitel „Optimierung unter Unsicherheit — Monte-Carlo, Stochastik, Robustheit“","anker":"sec:loesungen-unsicherheit","text":"12.1 — Fluch des Durchschnitts. Weil sich Verzögerungen fortpflanzen und die Verteilung rechtsschief ist: Ein Gewerk, das früher fertig wird, beschleunigt selten den Gesamtablauf ("},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Warum jedes Projekt zu spät fertig wird","anker":"finde-den-denkfehler-warum-jedes-projekt-zu-spät-fertig-wird","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-unsicherheit","text":"1 — (b). Das kritische Verhältnis 1400/(1400+120) = 0{,}921 gibt an, wie weit man sich auf die günstigere Fehlerseite stellen soll: Bestellt wird das 92,1-%-Quantil des Bedarfs, hi"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-unsicherheit","text":""},{"seite":"anhang-loesungen.html","titel":"A.13 Lösungen zu Kapitel „Dynamische Programmierung — Die Bellman-Gleichung und Order-Execution“","anker":"sec:loesungen-dynamische-programmierung","text":"13.1 — Bausteine. Stufe = Tag 1–5; Zustand = verbleibende Distanz (ggf. plus Erschöpfungsgrad); Aktion = heutige Tagesetappe; Wertfunktion = minimale Restanstrengung. Nicht in den "},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Der Zustand, der zu wenig weiß","anker":"finde-den-denkfehler-der-zustand-der-zu-wenig-weiß","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-dynamische-programmierung","text":"1 — (b). V_t setzt V_{t+1} voraus: Eine Entscheidung lässt sich erst bewerten, wenn feststeht, was sie für die Zukunft bedeutet. Am Ende ist dieser Wert bekannt (V_T = g), deshalb "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-dynamische-programmierung","text":""},{"seite":"anhang-loesungen.html","titel":"A.14 Lösungen zu Kapitel „Mehrere Ziele — Pareto-Fronten statt Gewichte“","anker":"sec:loesungen-mehrziel","text":"14.1 — Dominanz prüfen. P (14 000 €, 6 000 kg) und Q (13 800 €, 6 100 kg): keiner dominiert den anderen. Q ist billiger, P ist sauberer — beide sind pareto-optimal, die Wahl ist ei"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Wir haben die Gewichte sauber kalibriert“","anker":"denkfehler-loesung-mehrziel","text":"Der Lösungsraum wurde nicht vollständig abgetastet — und zwar aus prinzipiellen Gründen."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-mehrziel","text":"1. b) Geometrisch heißt „gewichtete Summe minimieren“: eine Gerade von links unten an die Punktwolke schieben. Sie berührt immer einen Eckpunkt der unteren konvexen Hülle; Punkte d"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-mehrziel","text":""},{"seite":"anhang-loesungen.html","titel":"A.15 Lösungen zu Kapitel „Predict-then-Optimize“","anker":"sec:loesungen-prognose","text":"15.1 — Andere Preise. Der Überhang kostet nur noch 1,50 €, die Fehlmenge weiterhin 6 €. Das kritische Verhältnis steigt von 6/9 = 0{,}667 auf 6/7{,}5 = 0{,}80. Die Bestellmenge ver"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Wir haben die Prognose um 18 % verbessert“","anker":"denkfehler-loesung-prognose","text":"Der Sicherheitszuschlag stammt aus den Residuen des alten Modells."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-prognose","text":"1. b) Die Kostenasymmetrie verschiebt das Optimum vom Erwartungswert zum kritischen Quantil. (a) und (c) sind reale Probleme, aber nicht dieses: Selbst bei perfekt bekannter, norma"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-prognose","text":""},{"seite":"anhang-loesungen.html","titel":"A.16 Lösungen zu Kapitel „Die Strukturbrücke — dieselbe Mathematik, zwei Welten“","anker":"sec:loesungen-bruecke","text":"16.1 — Die dritte Ressource. „Liquidität” — wie schnell sich eine Position ohne Kursabschlag verkaufen lässt — entspricht in der Werkstatt der Umrüstbarkeit oder der Vorlaufzeit: W"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Das ist doch dasselbe Problem“","anker":"denkfehler-loesung-bruecke","text":"Fehler 1: Varianz ist hier das falsche Risikomaß — genau der Fall, für den es CVaR gibt."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-bruecke","text":"1. b) Beide Probleme haben dieselbe Struktur: knappe Größen auf konkurrierende Verwendungen verteilen. (a) ist falsch — Produktionsproblem ist streng typisiert und validiert beim E"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-bruecke","text":""},{"seite":"anhang-loesungen.html","titel":"A.17 Lösungen zu Kapitel „Supply-Chain und Energieeinsatz unter Unsicherheit“","anker":"sec:loesungen-supplychain","text":"17.1 — Der Umschlagpunkt. Bei 60 MW statt 120 MW halbiert sich die Grenzkostendifferenz je Stunde (83 €/MWh × 60 MW = 4 980 €/h). Der Umschlagpunkt verdoppelt sich auf 7,7 Stunden."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Wir rechnen mit dem P50-Szenario“","anker":"denkfehler-loesung-supplychain","text":"Fehler 1: Der Median ist keine Absicherung, sondern eine Wette auf die Hälfte der Fälle."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-supplychain","text":"1. b) Anfahrkosten und Mindestlaufzeiten koppeln die Stunden. Die Merit-Order ist eine Sortierung und kennt keine Kopplung über die Zeit; sie ist für einen einzelnen Zeitpunkt rich"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-supplychain","text":""},{"seite":"anhang-loesungen.html","titel":"A.18 Lösungen zu Kapitel „Finanzdaten-Modellierung — Renditen, Kovarianz und Shrinkage“","anker":"sec:loesungen-finanzdaten","text":"18.1 — Renditeart. (a) diskret. (b) logarithmisch. (c) logarithmisch (bzw. diskret kumuliert über Produkt). (d) logarithmisch (statistische Eigenschaften)."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Das risikofreie Portfolio","anker":"finde-den-denkfehler-das-risikofreie-portfolio","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-finanzdaten","text":"1 — (b) bei 75 %. 1{,}0 \\cdot 0{,}5 \\cdot 1{,}5 = 0{,}75. Prozentuale Änderungen verknüpfen sich multiplikativ; ein Verlust wiegt schwerer als ein gleich großer Gewinn, weil er von"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-finanzdaten","text":""},{"seite":"anhang-loesungen.html","titel":"A.19 Lösungen zu Kapitel „Die moderne Portfoliotheorie nach Markowitz“","anker":"sec:loesungen-markowitz","text":"19.1 — Diversifikationseffekt."},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Zwölf gleiche Anlagen, ein sehr ungleiches Portfolio","anker":"finde-den-denkfehler-zwölf-gleiche-anlagen-ein-sehr-ungleiches-portfolio","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-markowitz","text":"1 — (b). Das renditestärkste Portfolio auf der Effizienzlinie hat notwendig das schlechteste Rendite-Risiko-Verhältnis (im Kapitelbeispiel 0,44 gegenüber 0,75 beim Optimum) und bes"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-markowitz","text":""},{"seite":"anhang-loesungen.html","titel":"A.20 Lösungen zu Kapitel „Tail-Risiko, CVaR und Transaktionskosten“","anker":"sec:loesungen-cvar","text":"20.1 — VaR und CVaR von Hand. Sortierte Verluste: -2{,}0; -1{,}2; -0{,}8; -0{,}4; 0{,}3; 0{,}6; 1{,}1; 2{,}1; 5{,}4; 8{,}9. Das 80 %-Quantil ist der 8. Wert: \\text{VaR}_{80\\%} = 2{"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Das Risikobudget, das durch Diversifikation stieg","anker":"finde-den-denkfehler-das-risikobudget-das-durch-diversifikation-stieg","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-cvar","text":"1 — (b) nichts. Der VaR ist ein Quantil: Er markiert die Schwelle und sagt nichts über den Bereich dahinter. Genau das zeigt der Schnellstart dieses Kapitels — zwei Anlagen mit ide"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-cvar","text":""},{"seite":"anhang-loesungen.html","titel":"A.21 Lösungen zu Kapitel „Die vollständige quantitative Handelsmaschine“","anker":"sec:loesungen-handelsmaschine","text":"21.1 — Lookahead erkennen. (a) sauber (:heute). (b) Lookahead — gesamter Zeitraum. (c) Lookahead — Volatilität über den ganzen Zeitraum enthält Zukunft. (d) sauber — Gewichte aus D"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Die Strategie mit dem hochsignifikanten Ergebnis","anker":"finde-den-denkfehler-die-strategie-mit-dem-hochsignifikanten-ergebnis","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-handelsmaschine","text":"1 — (b) nichts. Beide Kennzahlen sind Mittelwerte über die Zeit und blind für die Reihenfolge. Der Schnellstart dieses Kapitels zeigt es an denselben Tagesrenditen, nur anders ange"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-handelsmaschine","text":""},{"seite":"anhang-loesungen.html","titel":"A.22 Lösungen zu Kapitel „Praxisfallen und der Weg zum produktiven Einsatz“","anker":"sec:loesungen-praxisfallen","text":"22.1 — Hart oder weich, revisited. (a) hart: gesetzliche Ruhezeit, Qualifikationspflicht. (b) weich: individuelle Wunschtage, Vermeidung von Freistunden. (c) diskutabel: Höchstzahl"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — Das Modell, das seit einem Jahr nicht mehr optimiert","anker":"finde-den-denkfehler-das-modell-das-seit-einem-jahr-nicht-mehr-optimiert","text":""},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-praxisfallen","text":"1 — (b) das Zeitlimit greift. Eine Laufzeit, die exakt dem Limit entspricht und über Monate konstant bleibt, ist kein Stabilitätsbeleg, sondern zeigt, dass der Solver jedes Mal abg"},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-praxisfallen","text":""},{"seite":"anhang-loesungen.html","titel":"A.23 Lösungen zu Kapitel „Testen, Messen, Ausliefern“","anker":"sec:loesungen-testing","text":"23.1 — Den Schnellstart reparieren. Zwei Wege: (a) Abrunden statt runden — np.floor statt np.round. Das ist immer zulässig, weil weniger produzieren nie eine Kapazität sprengt, kos"},{"seite":"anhang-loesungen.html","titel":"Finde den Denkfehler — „Die Suite ist grün, das Modell stimmt”","anker":"denkfehler-loesung-testing","text":"Alle Prüfungen benutzen dieselbe falsche Zahl."},{"seite":"anhang-loesungen.html","titel":"Micro-Quiz","anker":"quiz-loesung-testing","text":"1. b) Es gibt keine unabhängige Quelle für die richtige Antwort — sonst bräuchte man den Solver nicht. (a) ist bei festgelegtem Seed und Zeitlimit meist beherrschbar; (c) löst man "},{"seite":"anhang-loesungen.html","titel":"Selbsttest","anker":"selbsttest-loesung-testing","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"Anhang B: Katalog der Modellierungsmuster","anker":null,"text":""},{"seite":"anhang-modellierungsmuster.html","titel":"Übersicht","anker":"übersicht","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"Logische Schalter","anker":"logische-schalter","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B1 — Aktivierungsschalter (Fixkosten)","anker":"b1-aktivierungsschalter-fixkosten","text":"Regel. „Wenn überhaupt etwas produziert wird, fallen Rüstkosten F an.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B2 — Semikontinuierliche Variable","anker":"b2-semikontinuierliche-variable","text":"Regel. „Entweder gar nicht oder mindestens L (und höchstens U).“"},{"seite":"anhang-modellierungsmuster.html","titel":"B3 — Implikation","anker":"b3-implikation","text":"Regel. „Wenn A gewählt wird, muss auch B gewählt werden.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B4 — Entweder-Oder (disjunktive Bedingung)","anker":"b4-entweder-oder-disjunktive-bedingung","text":"Regel. „Es muss f(\\mathbf{x}) \\le b_1 oder g(\\mathbf{x}) \\le b_2 gelten.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B5 — Exklusiv-Oder","anker":"b5-exklusiv-oder","text":"Regel. „Genau eine der Optionen wird gewählt.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B6 — Kardinalität","anker":"b6-kardinalität","text":"Regel. „Höchstens (mindestens, genau) K von N.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B7 — Bedingte Kopplung (Konjunktion)","anker":"b7-bedingte-kopplung-konjunktion","text":"Regel. „Wenn A und B, dann auch C.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Mengen und Grenzen","anker":"mengen-und-grenzen","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B8 — Weiche Grenze mit Strafkosten","anker":"b8-weiche-grenze-mit-strafkosten","text":"Regel. „Möglichst nicht über b — wenn doch, kostet es.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B9 — Gestaffelte Preise (stückweise linear)","anker":"b9-gestaffelte-preise-stückweise-linear","text":"Regel. „Die ersten 100 Stück kosten 5 €, danach 8 €.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B10 — Absolutbetrag / Abweichung","anker":"b10-absolutbetrag-abweichung","text":"Regel. „Die Abweichung vom Zielwert soll klein sein, egal in welche Richtung.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B11 — Min/Max in der Zielfunktion","anker":"b11-minmax-in-der-zielfunktion","text":"Regel. „Der am schlechtesten gestellte Beteiligte soll möglichst gut dastehen“ (Maximin/Fairness)."},{"seite":"anhang-modellierungsmuster.html","titel":"B12 — Verhältnis-Bedingung","anker":"b12-verhältnis-bedingung","text":"Regel. „Der Anteil von Gruppe G soll mindestens 30 % betragen.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B25 — Mindestabnahmemenge über einen Zeitraum","anker":"b25-mindestabnahmemenge-über-einen-zeitraum","text":"Regel. „Entweder wir arbeiten mit diesem Lieferanten gar nicht, oder wir nehmen ihm im Jahr mindestens 500 Einheiten ab.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B27 — Budgetlimit","anker":"b27-budgetlimit","text":"Regel. „Alle Maßnahmen zusammen dürfen 2 Mio. € nicht überschreiten.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Zeit und Reihenfolge","anker":"zeit-und-reihenfolge","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B13 — Vorrangbeziehung","anker":"b13-vorrangbeziehung","text":"Regel. „Arbeitsgang B darf erst beginnen, wenn A fertig ist.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B14 — Nichtüberlappung","anker":"b14-nichtüberlappung","text":"Regel. „Eine Maschine bearbeitet nur einen Job gleichzeitig.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B15 — Kumulative Ressource","anker":"b15-kumulative-ressource","text":"Regel. „Zu keinem Zeitpunkt dürfen mehr als 3 Arbeiten gleichzeitig laufen“ bzw. „die Stromlast darf 500 kW nie übersteigen“."},{"seite":"anhang-modellierungsmuster.html","titel":"B16 — Gleitendes Fenster","anker":"b16-gleitendes-fenster","text":"Regel. „Höchstens 5 Arbeitstage in Folge“ / „mindestens 2 freie Tage je Woche“."},{"seite":"anhang-modellierungsmuster.html","titel":"B17 — Umrüst- bzw. Wechselkosten","anker":"b17-umrüst--bzw.-wechselkosten","text":"Regel. „Ein Produktwechsel kostet Rüstzeit.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B26 — Rüstzeit als Kapazitätsverbrauch","anker":"b26-rüstzeit-als-kapazitätsverbrauch","text":"Regel. „Das Umrüsten kostet nicht nur Geld, es kostet Maschinenstunden — und die fehlen dann für die Produktion.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Robustheit und Diagnose","anker":"robustheit-und-diagnose","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B18 — Schlupfvariablen gegen Unlösbarkeit","anker":"b18-schlupfvariablen-gegen-unlösbarkeit","text":"Das wichtigste Muster für den Produktivbetrieb."},{"seite":"anhang-modellierungsmuster.html","titel":"B19 — Hierarchische Ziele (lexikografisch)","anker":"b19-hierarchische-ziele-lexikografisch","text":"Regel. „Erst die Besetzung sicherstellen, dann die Fairness optimieren.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B20 — Symmetriebrechung","anker":"b20-symmetriebrechung","text":"Regel. „Drei identische Maschinen — der Solver soll nicht alle Vertauschungen durchprobieren.“"},{"seite":"anhang-modellierungsmuster.html","titel":"B21 — Worst-Case-Abzug (robuste Formulierung)","anker":"b21-worst-case-abzug-robuste-formulierung","text":"Regel. „Rechne nicht mit dem geschätzten Wert, sondern mit dem ungünstigsten plausiblen.“"},{"seite":"anhang-modellierungsmuster.html","titel":"Netzwerke","anker":"netzwerke","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B22 — Flusserhaltung","anker":"b22-flusserhaltung","text":"\\sum_{j:(i,j)\\in E} x_{ij} - \\sum_{k:(k,i)\\in E} x_{ki} = b_i \\qquad \\forall i"},{"seite":"anhang-modellierungsmuster.html","titel":"B23 — Zuordnung 1:1","anker":"b23-zuordnung-11","text":""},{"seite":"anhang-modellierungsmuster.html","titel":"B24 — Subtour-Eliminierung","anker":"b24-subtour-eliminierung","text":"MTZ (einfach, aber schwach): u_i - u_j + C x_{ij} \\le C - d_j \\qquad \\forall i\\ne j"},{"seite":"anhang-modellierungsmuster.html","titel":"Ein Wort zur Auswahl","anker":"ein-wort-zur-auswahl","text":"Wenn mehrere Muster passen, entscheiden Sie nach dieser Reihenfolge:"},{"seite":"anhang-fehlerdiagnose.html","titel":"Anhang C: Fehlerdiagnose-Handbuch","anker":null,"text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"Symptom-Schnellübersicht","anker":"symptom-schnellübersicht","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C1 — INFEASIBLE","anker":"c1-infeasible","text":"Bedeutung. Es gibt keinen einzigen Punkt, der alle Bedingungen gleichzeitig erfüllt. Das ist eine Aussage über Ihr Modell, nicht über den Solver."},{"seite":"anhang-fehlerdiagnose.html","titel":"Diagnose in fünf Schritten","anker":"diagnose-in-fünf-schritten","text":"Schritt 1 — Trivialprüfungen von Hand. Rechnen Sie die offensichtlichen Bilanzen nach, bevor Sie den Solver befragen:"},{"seite":"anhang-fehlerdiagnose.html","titel":"Wenn die fünf Schritte nicht reichen: den Konflikt einkreisen","anker":"wenn-die-fünf-schritte-nicht-reichen-den-konflikt-einkreisen","text":"Warum Schritt 2 so oft ins Leere läuft. Bedingungen einzeln abzuschalten funktioniert nur, solange es einen Schuldigen gibt. Überlagern sich zwei unabhängige Widersprüche, bleibt d"},{"seite":"anhang-fehlerdiagnose.html","titel":"C2 — UNBOUNDED","anker":"c2-unbounded","text":"Bedeutung. Die Zielfunktion lässt sich unbegrenzt verbessern."},{"seite":"anhang-fehlerdiagnose.html","titel":"C3 — Zu langsam","anker":"c3-zu-langsam","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"Diagnosereihenfolge","anker":"diagnosereihenfolge","text":"1. Ist es überhaupt das Lösen?"},{"seite":"anhang-fehlerdiagnose.html","titel":"C4 — Unsinniges Ergebnis","anker":"c4-unsinniges-ergebnis","text":"Der Solver hat immer recht — bezogen auf das Modell, das Sie ihm gegeben haben. Wenn das Ergebnis unsinnig ist, beschreibt Ihr Modell nicht das Problem, das Sie meinen."},{"seite":"anhang-fehlerdiagnose.html","titel":"Checkliste","anker":"checkliste","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"Die wirksamste Gegenmaßnahme","anker":"die-wirksamste-gegenmaßnahme","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C5 — Instabile Lösung","anker":"c5-instabile-lösung","text":"Symptom. Ein zusätzlicher Handelstag, und das Portfolio sieht völlig anders aus."},{"seite":"anhang-fehlerdiagnose.html","titel":"Derselbe Fehler außerhalb der Finanzwelt","anker":"derselbe-fehler-außerhalb-der-finanzwelt","text":"Instabilität ist kein Portfolioproblem, sondern ein Skalierungsproblem — und es trifft jedes Modell, in dem Größen sehr verschiedener Ordnung nebeneinanderstehen. Euro-Beträge (10^"},{"seite":"anhang-fehlerdiagnose.html","titel":"C6 — Falsche Dualwerte","anker":"c6-falsche-dualwerte","text":"Symptom. Schattenpreise sind negativ, wo sie positiv sein sollten — oder null bei einer offensichtlich knappen Ressource."},{"seite":"anhang-fehlerdiagnose.html","titel":"Die drei Prüfungen","anker":"die-drei-prüfungen","text":"1. Vorzeichenkonvention. Haben Sie zur Maximierung negiert?"},{"seite":"anhang-fehlerdiagnose.html","titel":"C7 — Widersprüchliche Solver","anker":"c7-widersprüchliche-solver","text":"Symptom. linprog sagt 530, CVXPY sagt 529,8."},{"seite":"anhang-fehlerdiagnose.html","titel":"C8 — DCPError","anker":"c8-dcperror","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C9 — Importfehler","anker":"c9-importfehler","text":""},{"seite":"anhang-fehlerdiagnose.html","titel":"C10 — Verdächtig guter Backtest","anker":"c10-verdächtig-guter-backtest","text":"Faustregel: Eine Sharpe Ratio über 2 bei einer einfachen Strategie ist fast immer ein Fehler, kein Fund."},{"seite":"anhang-fehlerdiagnose.html","titel":"Prüfreihenfolge","anker":"prüfreihenfolge","text":"1. Lookahead-Selbsttest."},{"seite":"anhang-fehlerdiagnose.html","titel":"C11 — Vertauschte Spalten","anker":"c11-vertauschte-spalten","text":"Symptom. Ergebnisse sind plausibel, aber falsch beschriftet — oder eine Sektorgrenze greift auf die falschen Titel."},{"seite":"anhang-fehlerdiagnose.html","titel":"Die goldene Diagnoseregel","anker":"die-goldene-diagnoseregel","text":""},{"seite":"anhang-spickzettel.html","titel":"Anhang D: Spickzettel der Solver","anker":null,"text":""},{"seite":"anhang-spickzettel.html","titel":"D1 — SciPy: linprog und milp","anker":"d1-scipy-linprog-und-milp","text":"Wofür. Die Einstiegsschicht: keine zusätzliche Installation, HiGHS als Unterbau, ideal für lineare und gemischt-ganzzahlige Probleme in Matrixform. Wofür nicht: alles, was sich nic"},{"seite":"anhang-spickzettel.html","titel":"Lineares Programm","anker":"lineares-programm","text":""},{"seite":"anhang-spickzettel.html","titel":"Ganzzahlig: milp","anker":"ganzzahlig-milp","text":""},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine","text":""},{"seite":"anhang-spickzettel.html","titel":"D2 — HiGHS über highspy","anker":"d2-highs-über-highspy","text":"Wofür. Derselbe Solver wie unter SciPy, aber direkt gesteuert: Optionen, Warm-Starts, inkrementelles Ändern eines bestehenden Modells. Wofür nicht: schnelles Hinschreiben — die CSR"},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-1","text":""},{"seite":"anhang-spickzettel.html","titel":"D3 — OR-Tools: pywraplp","anker":"d3-or-tools-pywraplp","text":"Wofür. Bequeme algebraische Schreibweise für LP und MILP mit umschaltbarem Backend (GLOP, SCIP, CBC, SAT). Wofür nicht: Scheduling und kombinatorische Bedingungen — dafür ist CP-SA"},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-2","text":""},{"seite":"anhang-spickzettel.html","titel":"D4 — CP-SAT: cp_model","anker":"d4-cp-sat-cp_model","text":"Wofür. Scheduling, Zuordnung, Reihenfolgen, alles Kombinatorische mit globalen Bedingungen. Wofür nicht: stetige Größen — CP-SAT rechnet ausschließlich ganzzahlig. Wer Nachkommaste"},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-3","text":""},{"seite":"anhang-spickzettel.html","titel":"D5 — CVXPY","anker":"d5-cvxpy","text":"Wofür. Konvexe Probleme: quadratische Ziele, Normen, CVaR, alles mit Regularisierungstermen. Wofür nicht: große kombinatorische Modelle — der Aufbau der Ausdrücke wird dann selbst "},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-4","text":""},{"seite":"anhang-spickzettel.html","titel":"D6 — Modellierungssprachen: Pyomo und Linopy","anker":"d6-modellierungssprachen-pyomo-und-linopy","text":"Wofür. Beide trennen Modell von Solver: dasselbe Modell läuft ohne Änderung unter HiGHS, CBC, Gurobi. Pyomo denkt in Mengen und Indizes wie eine mathematische Formulierung; Linopy "},{"seite":"anhang-spickzettel.html","titel":"Pyomo","anker":"pyomo","text":""},{"seite":"anhang-spickzettel.html","titel":"Linopy","anker":"linopy","text":""},{"seite":"anhang-spickzettel.html","titel":"Die drei häufigsten Stolpersteine","anker":"die-drei-häufigsten-stolpersteine-5","text":""},{"seite":"anhang-spickzettel.html","titel":"Die gemeinsame Regel","anker":"die-gemeinsame-regel","text":"Alle sechs Seiten haben denselben längsten Abschnitt: die Statusauswertung. Das ist kein Zufall. Ein Solveraufruf hat nie zwei Ausgänge, sondern mindestens fünf — optimal, zulässig"},{"seite":"anhang-glossar.html","titel":"Anhang E: Glossar","anker":null,"text":""},{"seite":"anhang-literatur.html","titel":"Anhang F: Literaturverzeichnis","anker":null,"text":""},{"seite":"anhang-literatur.html","titel":"Grundlagen und Standardwerke des Operations Research","anker":"grundlagen-und-standardwerke-des-operations-research","text":"Zu Kapitel 1 und Kapitel 2."},{"seite":"anhang-literatur.html","titel":"Modellierung in Python: Bibliotheken und Modellierungsschichten","anker":"modellierung-in-python-bibliotheken-und-modellierungsschichten","text":"Zu Kapitel 3."},{"seite":"anhang-literatur.html","titel":"Lineare und ganzzahlige Optimierung","anker":"lineare-und-ganzzahlige-optimierung","text":"Zu Kapitel 5 und Kapitel 6."},{"seite":"anhang-literatur.html","titel":"Moderne Solver: Implementierung, Vergleich und Benchmarks","anker":"moderne-solver-implementierung-vergleich-und-benchmarks","text":"Zu Kapitel 3 und Anhang D. Neuer Abschnitt: Das Buch vergleicht durchgehend Solver, die Literatur dazu stand bisher verstreut."},{"seite":"anhang-literatur.html","titel":"Constraint Programming und Scheduling","anker":"constraint-programming-und-scheduling","text":"Zu Kapitel 7."},{"seite":"anhang-literatur.html","titel":"Graphen, Flüsse und Tourenplanung","anker":"graphen-flüsse-und-tourenplanung","text":"Zu Kapitel 8. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Metaheuristiken","anker":"metaheuristiken","text":"Zu Kapitel 9. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Spaltengenerierung und Dekomposition","anker":"spaltengenerierung-und-dekomposition","text":"Zu Kapitel 10. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Konvexe und nichtlineare Optimierung","anker":"konvexe-und-nichtlineare-optimierung","text":"Zu Kapitel 11."},{"seite":"anhang-literatur.html","titel":"Optimierung unter Unsicherheit: stochastisch, robust, Chance Constraints","anker":"optimierung-unter-unsicherheit-stochastisch-robust-chance-constraints","text":"Zu Kapitel 12 und Kapitel 17. Aus dem früheren Block „Konvexe, nichtlineare und robuste Optimierung” herausgelöst und erweitert."},{"seite":"anhang-literatur.html","titel":"Dynamische Programmierung und sequenzielle Entscheidungen","anker":"dynamische-programmierung-und-sequenzielle-entscheidungen","text":"Zu Kapitel 13. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Mehrkriterielle Optimierung","anker":"mehrkriterielle-optimierung","text":"Zu Kapitel 14. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Lernbasierte Optimierung: maschinelles Lernen und OR","anker":"lernbasierte-optimierung-maschinelles-lernen-und-or","text":"Zu Kapitel 15. Neuer Abschnitt — im Buch als eigenes Kapitel behandelt, in der Literaturliste bisher gar nicht vertreten."},{"seite":"anhang-literatur.html","titel":"Supply Chain, Energie und Produktionsplanung","anker":"supply-chain-energie-und-produktionsplanung","text":"Zu Kapitel 16 und Kapitel 17. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Quantitative Finanzmathematik","anker":"quantitative-finanzmathematik","text":"Zu Kapitel 18, Kapitel 19, Kapitel 20 und Kapitel 21."},{"seite":"anhang-literatur.html","titel":"Praxis: Betrieb, Test und Auslieferung von Optimierungsmodellen","anker":"praxis-betrieb-test-und-auslieferung-von-optimierungsmodellen","text":"Zu Kapitel 22 und Kapitel 23. Neuer Abschnitt."},{"seite":"anhang-literatur.html","titel":"Software, Dokumentation und Datenquellen","anker":"software-dokumentation-und-datenquellen","text":""},{"seite":"anhang-literatur.html","titel":"Verbände, Normen und Regulierung","anker":"verbände-normen-und-regulierung","text":""},{"seite":"anhang-literatur.html","titel":"Änderungsprotokoll","anker":"änderungsprotokoll","text":"Stand: September 2026. Überarbeitung des Literaturverzeichnisses gegenüber der Vorgängerfassung."},{"seite":"anhang-literatur.html","titel":"Prüfung gegen Crossref","anker":"prüfung-gegen-crossref","text":"Alle Einträge wurden einzeln gegen die Crossref-API geprüft — bei einem Literaturverzeichnis ist eine erfundene oder falsch zitierte Quelle der schwerste Fehler, weil der Leser sie"}];
diff --git a/OR_HTML_04/gesamtdokument.html b/OR_HTML_04/gesamtdokument.html
index d88d4ea..4be9041 100644
--- a/OR_HTML_04/gesamtdokument.html
+++ b/OR_HTML_04/gesamtdokument.html
@@ -388,6 +388,7 @@
 <li><a href="#über-den-kurs" id="toc-über-den-kurs">Über den Kurs</a>
 <ul>
 <li><a href="#vorwort" id="toc-vorwort">Vorwort</a></li>
+<li><a href="#die-erstellung-der-quelldateien-programmbeispiele-des-glossars-stichwortverzeichnisses-sowie-der-online--und-pdf-fassung-erfolgte-in-einer-linux-basierten-arbeitsumgebung.-dabei-kamen-unter-anderem-textbasierte-auszeichnungsformate-und-werkzeuge-wie-markdown-pandoc-und-latex-zum-einsatz-auch-bei-einzelnen-technischen-und-redaktionellen-arbeitsschritten-wurden-ki-systeme-unterstützend-verwendet." id="toc-die-erstellung-der-quelldateien-programmbeispiele-des-glossars-stichwortverzeichnisses-sowie-der-online--und-pdf-fassung-erfolgte-in-einer-linux-basierten-arbeitsumgebung.-dabei-kamen-unter-anderem-textbasierte-auszeichnungsformate-und-werkzeuge-wie-markdown-pandoc-und-latex-zum-einsatz-auch-bei-einzelnen-technischen-und-redaktionellen-arbeitsschritten-wurden-ki-systeme-unterstützend-verwendet.">Die Erstellung der Quelldateien, Programmbeispiele, des Glossars, Stichwortverzeichnisses sowie der Online- und PDF-Fassung erfolgte in einer Linux-basierten Arbeitsumgebung. Dabei kamen unter anderem textbasierte Auszeichnungsformate und Werkzeuge wie Markdown, Pandoc und LaTeX zum Einsatz; auch bei einzelnen technischen und redaktionellen Arbeitsschritten wurden KI-Systeme unterstützend verwendet.</a></li>
 <li><a href="#ziel-der-masterclass" id="toc-ziel-der-masterclass">Ziel der Masterclass</a>
 <ul>
 <li><a href="#vorausgesetzte-kenntnisse" id="toc-vorausgesetzte-kenntnisse">Vorausgesetzte Kenntnisse</a></li>
@@ -1369,7 +1370,9 @@
 <p><strong>Erstens: Formeln werden übersetzt.</strong> Zu jeder nicht-trivialen Formel gehört in dieser Ausgabe eine Lesehilfe, die jedes Symbol einzeln benennt, und eine Umschreibung in Alltagssprache. Formeln sind eine Abkürzung für Menschen, die den Inhalt schon kennen — wer ihn erst lernt, braucht den ausgeschriebenen Text daneben. Niemand sollte an der Notation scheitern, wenn er den Gedanken versteht.</p>
 <p><strong>Zweitens: Es wird geübt.</strong> Über 130 Aufgaben mit vollständigen Lösungen begleiten die Kapitel. Sie sind gestaffelt: von Verständnisfragen, die man im Kopf beantwortet, über Handrechnungen bis zu Programmieraufgaben, die auf den Kapitelbeispielen aufbauen. Wer Optimierung nur liest, kann anschließend über Optimierung reden. Wer die Aufgaben rechnet, kann optimieren.</p>
 <p><strong>Drittens: Der Weg zur eigenen Anwendung ist ausgeschildert.</strong> Die Projektwerkstatt am Ende enthält elf vollständig ausgearbeitete Projektaufträge — vom Schul-Vertretungsplaner über die Tourenoptimierung eines Lieferdienstes bis zum eigenen Portfolio-Rebalancer. Jeder Auftrag nennt Datenquellen, Modellskizze, Abnahmekriterien und Stolperfallen. Sie sind so zugeschnitten, dass sie in 10 bis 25 Stunden zu einem vorzeigbaren Ergebnis führen.</p>
-<hr />
+<p><strong>Hinweis zur Entstehung und zu verwendeten Werkzeugen</strong> Dieser Kurs wurde vom Autor konzipiert, redaktionell bearbeitet und technisch umgesetzt. Zur Unterstützung bei der Konzeption, sprachlichen Ausarbeitung und Überarbeitung einzelner Textpassagen wurden generative KI-Systeme eingesetzt.</p>
+<p>Die Verantwortung für Inhalt, Auswahl und Darstellung liegt beim Autor. Sämtliche fachlichen Aussagen, Herleitungen, Rechenwege, Python-Beispiele und Literaturangaben wurden überprüft. Die im Text angeführten Quellen wurden unabhängig recherchiert und verifiziert.</p>
+<h2 id="die-erstellung-der-quelldateien-programmbeispiele-des-glossars-stichwortverzeichnisses-sowie-der-online--und-pdf-fassung-erfolgte-in-einer-linux-basierten-arbeitsumgebung.-dabei-kamen-unter-anderem-textbasierte-auszeichnungsformate-und-werkzeuge-wie-markdown-pandoc-und-latex-zum-einsatz-auch-bei-einzelnen-technischen-und-redaktionellen-arbeitsschritten-wurden-ki-systeme-unterstützend-verwendet.">Die Erstellung der Quelldateien, Programmbeispiele, des Glossars, Stichwortverzeichnisses sowie der Online- und PDF-Fassung erfolgte in einer Linux-basierten Arbeitsumgebung. Dabei kamen unter anderem textbasierte Auszeichnungsformate und Werkzeuge wie Markdown, Pandoc und LaTeX zum Einsatz; auch bei einzelnen technischen und redaktionellen Arbeitsschritten wurden KI-Systeme unterstützend verwendet.</h2>
 <h2 class="unnumbered" id="ziel-der-masterclass">Ziel der Masterclass</h2>
 <p>Ein umfassendes, didaktisch von Grund auf aufgebautes Lehrbuch zur mathematischen Entscheidungsoptimierung (<strong>Operations Research, OR</strong>) mit praktischer Umsetzung in <strong>Python</strong>. Dieser Kurs der <strong>mathematischen Optimierung</strong> setzt kein Vorwissen in OR voraus und führt schrittweise von den algebraischen und kombinatorischen Grundlagen über Scheduling- und Constraint-Modelle bis hin zur professionellen Anwendung im quantitativen Aktienhandel, Portfoliomanagement und Risikocontrolling.</p>
 <h3 class="unnumbered" id="vorausgesetzte-kenntnisse">Vorausgesetzte Kenntnisse</h3>
diff --git a/OR_HTML_04/vorwort.html b/OR_HTML_04/vorwort.html
index 67ca296..18ddc8c 100644
--- a/OR_HTML_04/vorwort.html
+++ b/OR_HTML_04/vorwort.html
@@ -119,7 +119,9 @@
 <p><strong>Erstens: Formeln werden übersetzt.</strong> Zu jeder nicht-trivialen Formel gehört in dieser Ausgabe eine Lesehilfe, die jedes Symbol einzeln benennt, und eine Umschreibung in Alltagssprache. Formeln sind eine Abkürzung für Menschen, die den Inhalt schon kennen — wer ihn erst lernt, braucht den ausgeschriebenen Text daneben. Niemand sollte an der Notation scheitern, wenn er den Gedanken versteht.</p>
 <p><strong>Zweitens: Es wird geübt.</strong> Über 130 Aufgaben mit vollständigen Lösungen begleiten die Kapitel. Sie sind gestaffelt: von Verständnisfragen, die man im Kopf beantwortet, über Handrechnungen bis zu Programmieraufgaben, die auf den Kapitelbeispielen aufbauen. Wer Optimierung nur liest, kann anschließend über Optimierung reden. Wer die Aufgaben rechnet, kann optimieren.</p>
 <p><strong>Drittens: Der Weg zur eigenen Anwendung ist ausgeschildert.</strong> Die Projektwerkstatt am Ende enthält elf vollständig ausgearbeitete Projektaufträge — vom Schul-Vertretungsplaner über die Tourenoptimierung eines Lieferdienstes bis zum eigenen Portfolio-Rebalancer. Jeder Auftrag nennt Datenquellen, Modellskizze, Abnahmekriterien und Stolperfallen. Sie sind so zugeschnitten, dass sie in 10 bis 25 Stunden zu einem vorzeigbaren Ergebnis führen.</p>
-<hr />
+<p><strong>Hinweis zur Entstehung und zu verwendeten Werkzeugen</strong> Dieser Kurs wurde vom Autor konzipiert, redaktionell bearbeitet und technisch umgesetzt. Zur Unterstützung bei der Konzeption, sprachlichen Ausarbeitung und Überarbeitung einzelner Textpassagen wurden generative KI-Systeme eingesetzt.</p>
+<p>Die Verantwortung für Inhalt, Auswahl und Darstellung liegt beim Autor. Sämtliche fachlichen Aussagen, Herleitungen, Rechenwege, Python-Beispiele und Literaturangaben wurden überprüft. Die im Text angeführten Quellen wurden unabhängig recherchiert und verifiziert.</p>
+<h2 id="die-erstellung-der-quelldateien-programmbeispiele-des-glossars-stichwortverzeichnisses-sowie-der-online--und-pdf-fassung-erfolgte-in-einer-linux-basierten-arbeitsumgebung.-dabei-kamen-unter-anderem-textbasierte-auszeichnungsformate-und-werkzeuge-wie-markdown-pandoc-und-latex-zum-einsatz-auch-bei-einzelnen-technischen-und-redaktionellen-arbeitsschritten-wurden-ki-systeme-unterstützend-verwendet.">Die Erstellung der Quelldateien, Programmbeispiele, des Glossars, Stichwortverzeichnisses sowie der Online- und PDF-Fassung erfolgte in einer Linux-basierten Arbeitsumgebung. Dabei kamen unter anderem textbasierte Auszeichnungsformate und Werkzeuge wie Markdown, Pandoc und LaTeX zum Einsatz; auch bei einzelnen technischen und redaktionellen Arbeitsschritten wurden KI-Systeme unterstützend verwendet.</h2>
 <h2 class="unnumbered" id="ziel-der-masterclass">Ziel der Masterclass</h2>
 <p>Ein umfassendes, didaktisch von Grund auf aufgebautes Lehrbuch zur mathematischen Entscheidungsoptimierung (<strong>Operations Research, OR</strong>) mit praktischer Umsetzung in <strong>Python</strong>. Dieser Kurs der <strong>mathematischen Optimierung</strong> setzt kein Vorwissen in OR voraus und führt schrittweise von den algebraischen und kombinatorischen Grundlagen über Scheduling- und Constraint-Modelle bis hin zur professionellen Anwendung im quantitativen Aktienhandel, Portfoliomanagement und Risikocontrolling.</p>
 <h3 class="unnumbered" id="vorausgesetzte-kenntnisse">Vorausgesetzte Kenntnisse</h3>
diff --git a/Operations_Research_mit_Python_Version_04.md b/Operations_Research_mit_Python_Version_04.md
index 5aad961..37f5ff6 100644
--- a/Operations_Research_mit_Python_Version_04.md
+++ b/Operations_Research_mit_Python_Version_04.md
@@ -50,6 +50,11 @@ Dieses Kompendium versteht sich als **Arbeitsbuch**. Das zeigt sich an drei Stel
 
 **Drittens: Der Weg zur eigenen Anwendung ist ausgeschildert.** Die Projektwerkstatt am Ende enthält elf vollständig ausgearbeitete Projektaufträge — vom Schul-Vertretungsplaner über die Tourenoptimierung eines Lieferdienstes bis zum eigenen Portfolio-Rebalancer. Jeder Auftrag nennt Datenquellen, Modellskizze, Abnahmekriterien und Stolperfallen. Sie sind so zugeschnitten, dass sie in 10 bis 25 Stunden zu einem vorzeigbaren Ergebnis führen.
 
+**Hinweis zur Entstehung und zu verwendeten Werkzeugen** Dieser Kurs wurde vom Autor konzipiert, redaktionell bearbeitet und technisch umgesetzt. Zur Unterstützung bei der Konzeption, sprachlichen Ausarbeitung und Überarbeitung einzelner Textpassagen wurden generative KI-Systeme eingesetzt.
+
+Die Verantwortung für Inhalt, Auswahl und Darstellung liegt beim Autor. Sämtliche fachlichen Aussagen, Herleitungen, Rechenwege, Python-Beispiele und Literaturangaben wurden überprüft. Die im Text angeführten Quellen wurden unabhängig recherchiert und verifiziert.
+
+Die Erstellung der Quelldateien, Programmbeispiele, des Glossars, Stichwortverzeichnisses sowie der Online- und PDF-Fassung erfolgte in einer Linux-basierten Arbeitsumgebung. Dabei kamen unter anderem textbasierte Auszeichnungsformate und Werkzeuge wie Markdown, Pandoc und LaTeX zum Einsatz; auch bei einzelnen technischen und redaktionellen Arbeitsschritten wurden KI-Systeme unterstützend verwendet.
 ---
 
 ## Ziel der Masterclass {-}
diff --git a/Operations_Research_mit_Python_Version_04.pdf b/Operations_Research_mit_Python_Version_04.pdf
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diff --git a/Operations_Research_mit_Python_Version_04/00_Vorwort_und_Lesehilfe.md b/Operations_Research_mit_Python_Version_04/00_Vorwort_und_Lesehilfe.md
index cbae045..bbfdf34 100644
--- a/Operations_Research_mit_Python_Version_04/00_Vorwort_und_Lesehilfe.md
+++ b/Operations_Research_mit_Python_Version_04/00_Vorwort_und_Lesehilfe.md
@@ -59,6 +59,12 @@ die Tourenoptimierung eines Lieferdienstes bis zum eigenen Portfolio-Rebalancer.
 Auftrag nennt Datenquellen, Modellskizze, Abnahmekriterien und Stolperfallen. Sie sind so
 zugeschnitten, dass sie in 10 bis 25 Stunden zu einem vorzeigbaren Ergebnis führen.
 
+**Hinweis zur Entstehung und zu verwendeten Werkzeugen**
+Dieser Kurs wurde vom Autor konzipiert, redaktionell bearbeitet und technisch umgesetzt. Zur Unterstützung bei der Konzeption, sprachlichen Ausarbeitung und Überarbeitung einzelner Textpassagen wurden generative KI-Systeme eingesetzt.
+
+Die Verantwortung für Inhalt, Auswahl und Darstellung liegt beim Autor. Sämtliche fachlichen Aussagen, Herleitungen, Rechenwege, Python-Beispiele und Literaturangaben wurden überprüft. Die im Text angeführten Quellen wurden unabhängig recherchiert und verifiziert.
+
+Die Erstellung der Quelldateien, Programmbeispiele, des Glossars, Stichwortverzeichnisses sowie der Online- und PDF-Fassung erfolgte in einer Linux-basierten Arbeitsumgebung. Dabei kamen unter anderem textbasierte Auszeichnungsformate und Werkzeuge wie Markdown, Pandoc und LaTeX zum Einsatz; auch bei einzelnen technischen und redaktionellen Arbeitsschritten wurden KI-Systeme unterstützend verwendet.
 ---
 
 ## Ziel der Masterclass {-}