kleine Änderungen
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dschlueter
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195
bilder_04/erzeuge_3d_konvexitaet.py
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195
bilder_04/erzeuge_3d_konvexitaet.py
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#!/usr/bin/env python3
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# erzeuge_3d_konvexitaet.py
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"""
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Erzeugt die Landschaftsgrafik zum Abschnitt Konvexitaet:
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bilder_04/kap02_konvexitaet_landschaft.svg
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ERGAENZT das vorhandene kap02_konvexitaet.svg, ersetzt es nicht: Jenes zeigt die
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Sehnenbedingung - die Definition. Dieses zeigt ihre Folge - dass bei einer
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nicht-konvexen Funktion der Startpunkt darueber entscheidet, was am Ende
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herauskommt.
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Beide Tafeln zeigen DIESELBE Funktion wie das Kapitelprogramm
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Konvexitaet_Demo.py:
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links f(x) = x^2 + 3 sin(3x) - eindimensional
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rechts f(x1, x2) = x1^2 + 3 sin(3 x1) + x2^2 + 3 sin(3 x2) - separabel
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Die rechte Funktion ist die separable Fortsetzung der linken: Auf jeder Achse
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liegt exakt die Kurve der linken Tafel. Deshalb ist die rechte Tafel keine neue
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Behauptung, sondern dieselbe eine Dimension weiter - und die Taeler, die links
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nebeneinander liegen, werden rechts zu einem Gitter aus Mulden.
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Gerechnet wird mit BFGS aus denselben Startpunkten wie im Buchprogramm. Die
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fuenf Ergebnisse der linken Tafel muessen deshalb Ziffer fuer Ziffer der im
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Kapitel abgedruckten Tabelle entsprechen; pruefe_gegen_buch() bricht ab, wenn
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nicht. Ein Bild, das andere Minima zeigt als die Tabelle daneben, waere
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schlimmer als gar kein Bild.
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Aufruf (aus dem Repository-Wurzelverzeichnis):
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python3 bilder_04/erzeuge_3d_konvexitaet.py
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Benoetigt: numpy, scipy, matplotlib
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"""
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from __future__ import annotations
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import os
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import sys
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import numpy as np
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from scipy.optimize import minimize
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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from stil_04 import FARBEN, speichere # noqa: E402
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import matplotlib.pyplot as plt # noqa: E402
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# Die Startpunkte des Buchprogramms - nicht selbst gewaehlt.
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STARTPUNKTE = [-3.0, -1.5, 0.0, 1.5, 3.0]
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# Die im Kapitel abgedruckte Tabelle: Start -> (Minimum, Funktionswert).
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# Nur zum Gegenpruefen; gerechnet wird unabhaengig davon.
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TABELLE_IM_BUCH = {
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-3.0: (-2.4280, 3.3695),
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-1.5: (-0.4874, -2.7448),
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0.0: (-0.4874, -2.7448),
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1.5: (1.4606, -0.7042),
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3.0: (3.3817, 9.4570),
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}
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def welle(x):
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"""Die Funktion des Buchprogramms: Parabel mit aufmodulierter Welle."""
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return x ** 2 + 3.0 * np.sin(3.0 * x)
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def welle_1d(x):
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"""Dieselbe Funktion in der Form, die scipy.optimize erwartet."""
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return welle(x[0])
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def welle_2d(x):
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"""Separable Fortsetzung in zwei Dimensionen."""
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return welle(x[0]) + welle(x[1])
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def suche_1d():
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"""Lokale Suche von jedem Startpunkt - wie in Teil 2 des Buchprogramms."""
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ergebnisse = []
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for start in STARTPUNKTE:
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res = minimize(welle_1d, x0=[start], method="BFGS")
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ergebnisse.append((start, float(res.x[0]), float(res.fun)))
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return ergebnisse
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def suche_2d(start):
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"""Dieselbe lokale Suche in 2D, mit aufgezeichnetem Weg."""
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weg = [np.array(start, dtype=float)]
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minimize(welle_2d, x0=list(start), method="BFGS",
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callback=lambda xk: weg.append(np.array(xk, dtype=float)))
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return np.array(weg)
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def pruefe_gegen_buch(ergebnisse) -> None:
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"""Bild und abgedruckte Tabelle muessen dieselben Minima nennen."""
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for start, ort, wert in ergebnisse:
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soll_ort, soll_wert = TABELLE_IM_BUCH[start]
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if abs(ort - soll_ort) > 5e-4 or abs(wert - soll_wert) > 5e-4:
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raise SystemExit(
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f"Konvexitaets-Landschaft: Start {start:+.1f} fuehrt zu "
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f"x = {ort:.4f} (f = {wert:.4f}), das Kapitel druckt "
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f"x = {soll_ort:.4f} (f = {soll_wert:.4f}) ab.")
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gefunden = len({round(ort, 3) for _, ort, _ in ergebnisse})
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if gefunden != 4:
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raise SystemExit(
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f"Konvexitaets-Landschaft: {gefunden} verschiedene Minima, das "
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f"Kapitel druckt 4 ab.")
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def zeichne(ergebnisse, wege) -> None:
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figur, achsen = plt.subplots(1, 2, figsize=(10.0, 4.3))
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# --- links: die Funktion des Buchprogramms mit ihren fuenf Laeufen -----
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links = achsen[0]
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x = np.linspace(-3.8, 3.8, 800)
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links.plot(x, welle(x), "-", color=FARBEN["haupt"], linewidth=2.0,
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zorder=3)
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bester = min(ergebnisse, key=lambda z: z[2])
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for start, ort, wert in ergebnisse:
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global_ = abs(ort - bester[1]) < 1e-3
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farbe = FARBEN["gut"] if global_ else FARBEN["fehler"]
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links.annotate("", xy=(ort, wert), xytext=(start, welle(start)),
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arrowprops=dict(arrowstyle="->", color=farbe,
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linewidth=1.4,
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connectionstyle="arc3,rad=0.18"),
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zorder=4)
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links.plot(start, welle(start), "o", color=farbe, markersize=6,
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zorder=5)
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links.plot(ort, wert, "*", color=farbe, markersize=13, zorder=6)
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links.set_xlabel("$x$")
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links.set_ylabel("$f(x)$")
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links.set_title("fünf Startpunkte, vier verschiedene Minima",
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fontsize=10.5, color=FARBEN["text"])
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links.annotate("grün: im tiefsten Tal gelandet\nrot: in einem anderen",
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xy=(0.36, 0.97), xycoords="axes fraction", fontsize=8.5,
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color=FARBEN["text"], va="top")
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# --- rechts: dieselbe Funktion, eine Dimension weiter ------------------
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rechts = achsen[1]
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gitter = np.linspace(-3.8, 3.8, 420)
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xx, yy = np.meshgrid(gitter, gitter)
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zz = welle(xx) + welle(yy)
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rechts.contourf(xx, yy, zz, levels=28, cmap="Blues_r", alpha=0.7, zorder=1)
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rechts.contour(xx, yy, zz, levels=16, colors=[FARBEN["linie"]],
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linewidths=0.5, zorder=2)
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for name, weg, farbe, versatz, ausrichtung in wege:
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rechts.plot(weg[:, 0], weg[:, 1], "-", color=farbe, linewidth=2.2,
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zorder=4)
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rechts.plot(*weg[0], "o", color=farbe, markersize=7, zorder=5)
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rechts.plot(*weg[-1], "*", color=farbe, markersize=15, zorder=6)
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rechts.annotate(f"{name}\nf = {welle_2d(weg[-1]):.2f}",
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xy=weg[-1], xytext=versatz,
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textcoords="offset points", fontsize=8.5,
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color=farbe, fontweight="bold", ha=ausrichtung,
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zorder=7)
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rechts.set_xlabel("$x_1$")
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rechts.set_ylabel("$x_2$")
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rechts.set_title("dieselbe Funktion in zwei Dimensionen",
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fontsize=10.5, color=FARBEN["text"])
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rechts.grid(visible=False)
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speichere(figur, "kap02_konvexitaet_landschaft")
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if __name__ == "__main__":
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ergebnisse = suche_1d()
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for start, ort, wert in ergebnisse:
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print(f" Start {start:>5.1f} -> x = {ort:>8.4f} f = {wert:>8.4f}")
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pruefe_gegen_buch(ergebnisse)
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print(" stimmt mit der abgedruckten Tabelle überein (5 Zeilen, 4 Minima)")
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weg_a = suche_2d((-1.4, -1.6))
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weg_b = suche_2d((2.9, 3.1))
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if welle_2d(weg_a[-1]) >= welle_2d(weg_b[-1]):
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raise SystemExit("Konvexitaets-Landschaft: Lauf A landet nicht "
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"tiefer als Lauf B - Farben und Aussage verkehrt.")
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if np.linalg.norm(weg_a[-1] - weg_b[-1]) < 1.0:
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raise SystemExit("Konvexitaets-Landschaft: Beide 2D-Laeufe enden im "
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"selben Tal - dann ist keine Falle zu sehen.")
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# Der Begleittext nennt diesen Abstand ("mehr als 24 Einheiten").
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abstand = welle_2d(weg_b[-1]) - welle_2d(weg_a[-1])
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if not 24.0 <= abstand < 25.0:
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raise SystemExit(
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f"Konvexitaets-Landschaft: Der Abstand der beiden Laeufe betraegt "
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f"{abstand:.2f} Einheiten. Der Text im Kapitel sagt 'mehr als 24' "
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f"- beides muss zusammenpassen.")
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print(f" 2D: Lauf A endet bei f = {welle_2d(weg_a[-1]):.2f}, "
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f"Lauf B bei f = {welle_2d(weg_b[-1]):.2f}")
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# Versatz und Ausrichtung der Beschriftung: Lauf B endet in der oberen
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# rechten Ecke, dort muss der Text nach innen zeigen.
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zeichne(ergebnisse,
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[("Start A", weg_a, FARBEN["gut"], (16, 16), "left"),
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("Start B", weg_b, FARBEN["fehler"], (-14, -30), "right")])
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189
bilder_04/erzeuge_bnb_geometrie.py
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189
bilder_04/erzeuge_bnb_geometrie.py
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#!/usr/bin/env python3
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# erzeuge_bnb_geometrie.py
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"""
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Erzeugt die geometrische Tafel zum Branch-and-Bound im MILP-Kapitel:
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bilder_04/kap_milp_bnb_geometrie.svg
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ERGAENZT den Suchbaum kap_milp_suchbaum.svg, ersetzt ihn nicht: Der Baum zeigt,
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WELCHE Teilprobleme entstehen. Diese Tafel zeigt, WAS Verzweigen mit dem
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Loesungsraum macht - und warum es erlaubt ist.
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Links das Polyeder der LP-Relaxation mit seinem Optimum (3; 1,5). Rechts
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dasselbe Polyeder, aus dem der Streifen 1 < x2 < 2 herausgeschnitten ist; uebrig
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bleiben die Aeste A (x2 <= 1) und B (x2 >= 2) mit ihren eigenen Optima.
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Der Punkt, auf den es ankommt: In dem herausgeschnittenen Streifen liegt KEIN
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einziger ganzzahliger Punkt - es geht also nichts verloren. Der Streifen
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enthaelt aber das Optimum der Relaxation, und deshalb sinkt die Schranke von 21
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auf 20,67 bzw. 18. Verzweigen heisst, wertlosen Raum wegzuschneiden, bis der
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beste verbleibende Punkt ganzzahlig ist.
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Modell und LP-Loeser stammen aus erzeuge_branch_and_bound.py, damit Baum und
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Geometrie nicht auseinanderlaufen koennen. pruefe_befund() rechnet nach, dass im
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Streifen wirklich kein ganzzahliger Punkt liegt und dass die drei Zielwerte die
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des Baums sind.
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Aufruf (aus dem Repository-Wurzelverzeichnis):
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python3 bilder_04/erzeuge_bnb_geometrie.py
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Benoetigt: numpy, scipy, matplotlib
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"""
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from __future__ import annotations
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import itertools
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import os
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import sys
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import numpy as np
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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from stil_04 import FARBEN, speichere # noqa: E402
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import matplotlib.pyplot as plt # noqa: E402
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from erzeuge_branch_and_bound import A, B, C, loese # noqa: E402
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# Die Verzweigung, die der Baum als Erstes vornimmt: x2 <= 1 gegen x2 >= 2.
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GRENZE = 1.0
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# Was der Suchbaum ausweist; nur zum Gegenpruefen.
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SCHRANKEN_IM_BAUM = {"P0": 21.0, "A": 62.0 / 3.0, "B": 18.0}
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def gitter(masche: int = 420):
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x1 = np.linspace(0, 4.6, masche)
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x2 = np.linspace(0, 3.4, masche)
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return np.meshgrid(x1, x2)
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def zulaessig(xx1, xx2, unten=None, oben=None):
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"""Maske des zulaessigen Bereichs, wahlweise mit zusaetzlichen Schranken."""
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maske = np.ones_like(xx1, dtype=bool)
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for zeile, grenze in zip(A, B):
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maske &= (zeile[0] * xx1 + zeile[1] * xx2 <= grenze + 1e-9)
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if unten is not None:
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maske &= (xx2 >= unten - 1e-9)
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if oben is not None:
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maske &= (xx2 <= oben + 1e-9)
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return maske
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def ganzzahlige_punkte():
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"""Alle ganzzahligen zulaessigen Punkte - fuer die Tafel und den Test."""
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punkte = []
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for p1, p2 in itertools.product(range(0, 6), range(0, 5)):
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if np.all(A @ np.array([p1, p2], dtype=float) <= B + 1e-9):
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punkte.append((p1, p2))
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return punkte
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def pruefe_befund(loesungen, punkte) -> None:
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"""Die beiden Aussagen der Tafel muessen stimmen."""
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im_streifen = [p for p in punkte if GRENZE < p[1] < GRENZE + 1]
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if im_streifen:
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raise SystemExit(
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f"B&B-Geometrie: Im herausgeschnittenen Streifen liegen die "
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f"ganzzahligen Punkte {im_streifen}. Dann waere das Verzweigen "
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f"nicht verlustfrei, und die Tafel behauptete etwas Falsches.")
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for name, soll in SCHRANKEN_IM_BAUM.items():
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ist = loesungen[name][1]
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if abs(ist - soll) > 1e-6:
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raise SystemExit(
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f"B&B-Geometrie: {name} hat den Zielwert {ist:.4f}, der "
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f"Suchbaum weist {soll:.4f} aus.")
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def zahl(wert: float) -> str:
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return f"{wert:.2f}".rstrip("0").rstrip(".").replace(".", ",")
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def zeichne(loesungen, punkte) -> None:
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figur, achsen = plt.subplots(1, 2, figsize=(10.0, 4.5))
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xx1, xx2 = gitter()
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for achse in achsen:
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for p1, p2 in punkte:
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achse.plot(p1, p2, "o", color=FARBEN["gedaempft"], markersize=4.5,
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zorder=5)
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for zeile, grenze in zip(A, B):
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x = np.linspace(0, 4.6, 100)
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achse.plot(x, (grenze - zeile[0] * x) / zeile[1], "--",
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color=FARBEN["linie"], linewidth=1.1, zorder=3)
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achse.set_xlim(0, 4.6)
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achse.set_ylim(0, 3.4)
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achse.set_xlabel("$x_1$")
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achse.grid(linestyle=":", alpha=0.4)
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achsen[0].set_ylabel("$x_2$")
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# --- links: die Relaxation, ungeschnitten -----------------------------
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achsen[0].contourf(xx1, xx2, zulaessig(xx1, xx2).astype(float),
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levels=[0.5, 1.5], colors=[FARBEN["haupt"]],
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alpha=0.20, zorder=1)
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x_p0, z_p0 = loesungen["P0"]
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achsen[0].plot(*x_p0, "*", color=FARBEN["fehler"], markersize=17, zorder=6)
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achsen[0].annotate(f"LP-Optimum ({zahl(x_p0[0])}; {zahl(x_p0[1])})\n"
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f"Z = {zahl(z_p0)} — nicht ganzzahlig",
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xy=x_p0, xytext=(12, 12), textcoords="offset points",
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fontsize=8.5, color=FARBEN["fehler"],
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fontweight="bold", zorder=7)
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achsen[0].set_title("$P_0$ — die Relaxation", fontsize=10.5,
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color=FARBEN["text"])
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# --- rechts: derselbe Bereich, der Streifen heraus ---------------------
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for name, unten, oben, farbe, beschriftung in (
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("A", None, GRENZE, FARBEN["zweit"], f"Ast A: $x_2 \\leq 1$"),
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("B", GRENZE + 1, None, FARBEN["warnung"],
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f"Ast B: $x_2 \\geq 2$")):
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achsen[1].contourf(xx1, xx2,
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zulaessig(xx1, xx2, unten, oben).astype(float),
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levels=[0.5, 1.5], colors=[farbe], alpha=0.22,
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zorder=1)
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x_ast, z_ast = loesungen[name]
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achsen[1].plot(*x_ast, "*", color=farbe, markersize=15, zorder=6)
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# Ast A liegt an der Schnittkante; sein Text gehoert nach unten in die
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# eigene Flaeche, sonst laeuft er in den Streifen hinein.
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achsen[1].annotate(f"{beschriftung}\nZ = {zahl(z_ast)}",
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xy=x_ast,
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xytext=(-12, -30) if name == "A" else (12, 14),
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textcoords="offset points", fontsize=8.5,
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ha="right" if name == "A" else "left",
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color=farbe, fontweight="bold", zorder=7)
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achsen[1].axhspan(GRENZE, GRENZE + 1, color=FARBEN["fehler"], alpha=0.11,
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||||
zorder=2)
|
||||
achsen[1].axhline(GRENZE, color=FARBEN["fehler"], linewidth=1.3, zorder=4)
|
||||
achsen[1].axhline(GRENZE + 1, color=FARBEN["fehler"], linewidth=1.3,
|
||||
zorder=4)
|
||||
achsen[1].text(4.5, GRENZE + 0.5,
|
||||
"herausgeschnitten:\n$1 < x_2 < 2$\nkein ganzzahliger "
|
||||
"Punkt darin", fontsize=8, color=FARBEN["fehler"],
|
||||
ha="right", va="center", zorder=7)
|
||||
achsen[1].set_title("nach dem Verzweigen über $x_2$", fontsize=10.5,
|
||||
color=FARBEN["text"])
|
||||
|
||||
figur.suptitle(f"Verzweigen schneidet weg, wo kein ganzzahliger Punkt "
|
||||
f"liegt — die Schranke sinkt von {zahl(z_p0)} auf "
|
||||
f"{zahl(max(loesungen['A'][1], loesungen['B'][1]))}",
|
||||
fontsize=10.5, color=FARBEN["text"], y=1.02)
|
||||
speichere(figur, "kap_milp_bnb_geometrie")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
unendlich = float("inf")
|
||||
loesungen = {
|
||||
"P0": loese([0.0, 0.0], [unendlich, unendlich]),
|
||||
"A": loese([0.0, 0.0], [unendlich, GRENZE]),
|
||||
"B": loese([0.0, GRENZE + 1], [unendlich, unendlich]),
|
||||
}
|
||||
punkte = ganzzahlige_punkte()
|
||||
for name, (x, z) in loesungen.items():
|
||||
print(f" {name:<3} x = ({x[0]:.4f}; {x[1]:.4f}) Z = {z:.4f}")
|
||||
print(f" {len(punkte)} ganzzahlige zulässige Punkte")
|
||||
|
||||
pruefe_befund(loesungen, punkte)
|
||||
print(" Streifen 1 < x₂ < 2 enthält keinen ganzzahligen Punkt")
|
||||
print(" Zielwerte stimmen mit dem Suchbaum überein")
|
||||
|
||||
zeichne(loesungen, punkte)
|
||||
190
bilder_04/erzeuge_dualitaets_bruecke.py
Normal file
190
bilder_04/erzeuge_dualitaets_bruecke.py
Normal file
|
|
@ -0,0 +1,190 @@
|
|||
#!/usr/bin/env python3
|
||||
|
||||
# erzeuge_dualitaets_bruecke.py
|
||||
"""
|
||||
Erzeugt die Doppeltafel zum starken Dualitaetssatz:
|
||||
|
||||
bilder_04/kap_lp_dualitaets_bruecke.svg
|
||||
|
||||
ERGAENZT das vorhandene kap04_primal_dual.svg, ersetzt es nicht: Jenes stellt
|
||||
die beiden Probleme schematisch gegenueber - wer fragt was. Dieses zeigt ihre
|
||||
Geometrie und damit den Satz selbst.
|
||||
|
||||
Links der primale Mengenraum (x1, x2): ein beschraenktes Polyeder, das Maximum
|
||||
in einer Ecke. Rechts der duale Preisraum (y1, y2): ein nach oben offener
|
||||
Bereich, das Minimum ebenfalls in einer Ecke. Die beiden Bereiche haben nichts
|
||||
miteinander gemein - andere Achsen, andere Einheiten, der eine beschraenkt, der
|
||||
andere nicht. Trotzdem steht an beiden Optima dieselbe Zahl. Das ist der Satz,
|
||||
und das ist der einzige Grund fuer die Doppeltafel.
|
||||
|
||||
Der duale Raum hat drei Variablen; gezeichnet wird die Ebene y3 = 0. Das ist
|
||||
keine Vereinfachung, sondern folgt aus dem komplementaeren Schlupf: Das
|
||||
Marktlimit x1 <= 8 ist bei x1* = 7,5 nicht bindend, also ist y3* = 0. Der
|
||||
Selbsttest rechnet nach, dass das Minimum in dieser Ebene wirklich das Minimum
|
||||
des vollen Dualproblems ist - sonst zeigte die rechte Tafel einen zu grossen
|
||||
Wert und der optische Beweis waere keiner.
|
||||
|
||||
Modell und Loesung stammen aus dem Kapitelprogramm Dualitaet_Nachweis.py; hier
|
||||
wird nichts nachgebaut und nichts eingetragen.
|
||||
|
||||
Aufruf (aus dem Repository-Wurzelverzeichnis):
|
||||
python3 bilder_04/erzeuge_dualitaets_bruecke.py
|
||||
|
||||
Benoetigt: numpy, scipy, matplotlib
|
||||
"""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import itertools
|
||||
import os
|
||||
import sys
|
||||
|
||||
import numpy as np
|
||||
|
||||
HIER = os.path.dirname(os.path.abspath(__file__))
|
||||
BASIS = os.path.dirname(HIER)
|
||||
sys.path.insert(0, HIER)
|
||||
sys.path.insert(0, os.path.join(
|
||||
BASIS, "Operations_Research_mit_Python_Version_04_Programme"))
|
||||
|
||||
from stil_04 import FARBEN, speichere # noqa: E402
|
||||
import matplotlib.pyplot as plt # noqa: E402
|
||||
from Dualitaet_Nachweis import (A_PRIMAL, B_PRIMAL, # noqa: E402
|
||||
C_PRIMAL, loese)
|
||||
|
||||
|
||||
def ecken(A: np.ndarray, b: np.ndarray) -> np.ndarray:
|
||||
"""Ecken des Bereichs {x >= 0 : A x <= b}, gegen den Uhrzeigersinn.
|
||||
|
||||
Alle Paare von Randgeraden schneiden (die Achsen zaehlen mit), die
|
||||
unzulaessigen Schnittpunkte verwerfen, den Rest um den Schwerpunkt
|
||||
sortieren. Fuer zwei Dimensionen ist das vollstaendig."""
|
||||
zeilen = np.vstack([A, [[-1.0, 0.0], [0.0, -1.0]]])
|
||||
rechte = np.concatenate([b, [0.0, 0.0]])
|
||||
punkte = []
|
||||
for i, j in itertools.combinations(range(len(zeilen)), 2):
|
||||
M = zeilen[[i, j]]
|
||||
if abs(np.linalg.det(M)) < 1e-9:
|
||||
continue
|
||||
p = np.linalg.solve(M, rechte[[i, j]])
|
||||
if np.all(zeilen @ p <= rechte + 1e-9):
|
||||
punkte.append(p)
|
||||
punkte = np.unique(np.round(punkte, 9), axis=0)
|
||||
mitte = punkte.mean(axis=0)
|
||||
winkel = np.arctan2(punkte[:, 1] - mitte[1], punkte[:, 0] - mitte[0])
|
||||
return punkte[np.argsort(winkel)]
|
||||
|
||||
|
||||
def zeichne(x_stern, z_stern, y_stern, w_stern) -> None:
|
||||
figur, achsen = plt.subplots(1, 2, figsize=(10.2, 4.6))
|
||||
beschriftung = f"{z_stern:,.0f} €".replace(",", ".")
|
||||
|
||||
# --- links: der primale Mengenraum ------------------------------------
|
||||
links = achsen[0]
|
||||
polygon = ecken(A_PRIMAL, B_PRIMAL)
|
||||
links.fill(polygon[:, 0], polygon[:, 1], color=FARBEN["haupt"],
|
||||
alpha=0.18, zorder=2)
|
||||
links.plot(np.append(polygon[:, 0], polygon[0, 0]),
|
||||
np.append(polygon[:, 1], polygon[0, 1]), "-",
|
||||
color=FARBEN["haupt"], linewidth=1.8, zorder=3)
|
||||
|
||||
x1 = np.linspace(0, 11, 200)
|
||||
# Ein Niveau unter dem Optimum, das Optimum selbst, eines darueber - das
|
||||
# obere liegt sichtbar ausserhalb des Polyeders, und genau das ist die
|
||||
# Aussage: weiter geht es nicht.
|
||||
for niveau, stil in ((1500.0, ":"), (z_stern, "--"), (3000.0, ":")):
|
||||
# 150 x1 + 250 x2 = niveau
|
||||
links.plot(x1, (niveau - C_PRIMAL[0] * x1) / C_PRIMAL[1], stil,
|
||||
color=FARBEN["warnung"], linewidth=1.4, zorder=4)
|
||||
links.annotate("", xy=(4.2, 4.2), xytext=(2.4, 3.1),
|
||||
arrowprops=dict(arrowstyle="-|>", color=FARBEN["warnung"],
|
||||
linewidth=2.0), zorder=5)
|
||||
links.text(4.35, 4.25, r"$\mathbf{c} = (150, 250)^\top$", fontsize=8.5,
|
||||
color=FARBEN["warnung"])
|
||||
|
||||
links.plot(*x_stern, "*", color=FARBEN["gut"], markersize=17, zorder=6)
|
||||
links.annotate(f"$\\mathbf{{x}}^* = ({x_stern[0]:.1f}\\,;\\,"
|
||||
f"{x_stern[1]:.0f})$\n$Z^* = {beschriftung}$",
|
||||
xy=x_stern, xytext=(4.55, 1.05), textcoords="data",
|
||||
fontsize=9, color=FARBEN["gut"], fontweight="bold",
|
||||
ha="left", zorder=7,
|
||||
arrowprops=dict(arrowstyle="-", color=FARBEN["gut"],
|
||||
linewidth=1.0, alpha=0.7))
|
||||
links.set_xlim(0, 11)
|
||||
links.set_ylim(0, 9)
|
||||
links.set_xlabel("$x_1$ — Bots Typ A")
|
||||
links.set_ylabel("$x_2$ — Bots Typ B")
|
||||
links.set_title("primal: Mengen — maximieren, beschränkt",
|
||||
fontsize=10.5, color=FARBEN["text"])
|
||||
|
||||
# --- rechts: der duale Preisraum, Ebene y3 = 0 ------------------------
|
||||
rechts = achsen[1]
|
||||
g1 = np.linspace(0, 55, 320)
|
||||
g2 = np.linspace(0, 55, 320)
|
||||
yy1, yy2 = np.meshgrid(g1, g2)
|
||||
# A^T y >= c, hier ohne y3: 2 y1 + 4 y2 >= 150 und 5 y1 + 6 y2 >= 250
|
||||
zulaessig = ((A_PRIMAL[0, 0] * yy1 + A_PRIMAL[1, 0] * yy2
|
||||
>= C_PRIMAL[0] - 1e-9)
|
||||
& (A_PRIMAL[0, 1] * yy1 + A_PRIMAL[1, 1] * yy2
|
||||
>= C_PRIMAL[1] - 1e-9))
|
||||
rechts.contourf(yy1, yy2, zulaessig.astype(float), levels=[0.5, 1.5],
|
||||
colors=[FARBEN["zweit"]], alpha=0.18, zorder=2)
|
||||
for spalte, farbe in ((0, FARBEN["zweit"]), (1, FARBEN["zweit"])):
|
||||
# A[0, spalte] * y1 + A[1, spalte] * y2 = c[spalte]
|
||||
rechts.plot(g1, (C_PRIMAL[spalte] - A_PRIMAL[0, spalte] * g1)
|
||||
/ A_PRIMAL[1, spalte], "-", color=farbe, linewidth=1.8,
|
||||
zorder=3)
|
||||
|
||||
for niveau, stil in ((1200.0, ":"), (w_stern, "--"), (3600.0, ":")):
|
||||
# 40 y1 + 60 y2 = niveau
|
||||
rechts.plot(g1, (niveau - B_PRIMAL[0] * g1) / B_PRIMAL[1], stil,
|
||||
color=FARBEN["warnung"], linewidth=1.4, zorder=4)
|
||||
|
||||
rechts.plot(y_stern[0], y_stern[1], "*", color=FARBEN["gut"],
|
||||
markersize=17, zorder=6)
|
||||
rechts.annotate(f"$\\mathbf{{y}}^* = ({y_stern[0]:.2f}\\,;\\,"
|
||||
f"{y_stern[1]:.2f})$\n$W^* = "
|
||||
f"{f'{w_stern:,.0f}'.replace(',', '.')}$ €",
|
||||
xy=(y_stern[0], y_stern[1]), xytext=(14, 16),
|
||||
textcoords="offset points", fontsize=9,
|
||||
color=FARBEN["gut"], fontweight="bold", zorder=7)
|
||||
rechts.annotate("zulässig: nach oben offen —\nteurer geht immer",
|
||||
xy=(0.04, 0.97), xycoords="axes fraction", fontsize=8.5,
|
||||
color=FARBEN["text"], va="top")
|
||||
rechts.set_xlim(0, 55)
|
||||
rechts.set_ylim(0, 55)
|
||||
rechts.set_xlabel("$y_1$ — Preis je vCPU-Stunde")
|
||||
rechts.set_ylabel("$y_2$ — Preis je GB RAM")
|
||||
rechts.set_title("dual: Preise — minimieren, unbeschränkt",
|
||||
fontsize=10.5, color=FARBEN["text"])
|
||||
|
||||
figur.suptitle(f"Zwei verschiedene Räume, ein gemeinsamer Wert: "
|
||||
f"{beschriftung}",
|
||||
fontsize=11.5, color=FARBEN["text"], y=1.01)
|
||||
speichere(figur, "kap_lp_dualitaets_bruecke")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
primal, dual = loese()
|
||||
x_stern, z_stern = primal.x, -primal.fun
|
||||
y_stern, w_stern = dual.x, dual.fun
|
||||
print(f" primal: x* = ({x_stern[0]:.4f}, {x_stern[1]:.4f}) "
|
||||
f"Z* = {z_stern:.4f}")
|
||||
print(f" dual: y* = ({y_stern[0]:.4f}, {y_stern[1]:.4f}, "
|
||||
f"{y_stern[2]:.4f}) W* = {w_stern:.4f}")
|
||||
|
||||
if abs(z_stern - w_stern) > 1e-6:
|
||||
raise SystemExit(
|
||||
f"Dualitaets-Bruecke: Z* = {z_stern:.6f}, W* = {w_stern:.6f}. "
|
||||
f"Die Tafel behauptet einen gemeinsamen Wert - er ist keiner.")
|
||||
# Die rechte Tafel zeigt die Ebene y3 = 0. Das ist nur zulaessig, wenn
|
||||
# das Optimum dort liegt.
|
||||
if abs(y_stern[2]) > 1e-9:
|
||||
raise SystemExit(
|
||||
f"Dualitaets-Bruecke: y3* = {y_stern[2]:.6f} ist nicht null. Die "
|
||||
f"gezeichnete Ebene y3 = 0 enthaelt das Optimum dann nicht.")
|
||||
print(f" starker Dualitätssatz bestätigt: "
|
||||
f"{z_stern:,.0f} €".replace(",", "."))
|
||||
print(f" y3* = 0 — die gezeichnete Ebene enthält das Optimum")
|
||||
|
||||
zeichne(x_stern, z_stern, y_stern, w_stern)
|
||||
|
|
@ -91,9 +91,43 @@ def loese() -> tuple[list[tuple[int, int, int, int]], int]:
|
|||
return plan, loeser.Value(makespan)
|
||||
|
||||
|
||||
def untere_schranke() -> int:
|
||||
"""Wie in JobShop_Intervalle.py: laengster Auftrag bzw. hoechste Maschinenlast.
|
||||
|
||||
Kein Plan kann kuerzer sein als der laengste Auftrag (seine Schritte laufen
|
||||
nacheinander) und auch nicht kuerzer als die am staerksten belegte Maschine
|
||||
(ihre Schritte koennen sich nicht ueberlappen)."""
|
||||
je_auftrag = max(sum(dauer for _, dauer in auftrag)
|
||||
for auftrag in AUFTRAEGE)
|
||||
last = [0] * len(MASCHINENNAMEN)
|
||||
for auftrag in AUFTRAEGE:
|
||||
for maschine, dauer in auftrag:
|
||||
last[maschine] += dauer
|
||||
return max(je_auftrag, max(last))
|
||||
|
||||
|
||||
def stillstaende(plan):
|
||||
"""Luecken ZWISCHEN zwei Arbeitsgaengen derselben Maschine.
|
||||
|
||||
Nur die inneren: Was eine Maschine vor ihrem ersten oder nach ihrem letzten
|
||||
Arbeitsgang tut, ist keine Luecke im Plan, sondern sein Rand."""
|
||||
luecken = []
|
||||
for maschine in range(len(MASCHINENNAMEN)):
|
||||
zeiten = sorted((beginn, beginn + dauer)
|
||||
for _, m, beginn, dauer in plan if m == maschine)
|
||||
for (_, ende), (naechster_beginn, _) in zip(zeiten, zeiten[1:]):
|
||||
if naechster_beginn > ende:
|
||||
luecken.append((maschine, ende, naechster_beginn))
|
||||
return luecken
|
||||
|
||||
|
||||
def zeichne_statisch(plan, makespan: int) -> None:
|
||||
"""Matplotlib-Fassung fuer das PDF."""
|
||||
figur, achse = plt.subplots(figsize=(9.0, 3.2))
|
||||
figur, achse = plt.subplots(figsize=(9.0, 3.4))
|
||||
for maschine, von, bis in stillstaende(plan):
|
||||
achse.barh(y=maschine, width=bis - von, left=von, height=0.55,
|
||||
facecolor="none", edgecolor="#be123c", linewidth=1.2,
|
||||
hatch="///", zorder=2)
|
||||
for auftrag, maschine, beginn, dauer in plan:
|
||||
achse.barh(y=maschine, width=dauer, left=beginn, height=0.55,
|
||||
color=FARBEN[auftrag % len(FARBEN)],
|
||||
|
|
@ -108,7 +142,21 @@ def zeichne_statisch(plan, makespan: int) -> None:
|
|||
achse.set_xlabel("Zeiteinheiten")
|
||||
achse.set_xlim(0, makespan)
|
||||
achse.set_xticks(range(makespan + 1))
|
||||
achse.set_title(f"Job-Shop-Plan, Makespan = {makespan}", fontsize=11)
|
||||
|
||||
schranke = untere_schranke()
|
||||
achse.axvline(schranke, color="#059669", linestyle="--", linewidth=1.5,
|
||||
zorder=5)
|
||||
# x in Datenkoordinaten, y als Anteil der Achsenhoehe: Die y-Achse ist
|
||||
# umgekehrt, ein fester y-Wert laege sonst ausserhalb des Bildes.
|
||||
unten = achse.get_xaxis_transform()
|
||||
# Der Makespan steht schon im Titel - hier nur die Schranke, sonst
|
||||
# stossen beide Beschriftungen aneinander.
|
||||
achse.text(schranke - 0.12, 0.03, f"untere Schranke {schranke}",
|
||||
transform=unten, fontsize=8.5, color="#059669", ha="right",
|
||||
va="bottom", zorder=6)
|
||||
achse.set_title(f"Job-Shop-Plan: Makespan {makespan}, untere Schranke "
|
||||
f"{schranke} — schraffiert steht die Maschine still",
|
||||
fontsize=11)
|
||||
achse.grid(axis="x", linestyle=":", alpha=0.5)
|
||||
for rand in ("top", "right", "left"):
|
||||
achse.spines[rand].set_visible(False)
|
||||
|
|
@ -170,6 +218,40 @@ def zeichne_interaktiv(plan, makespan: int) -> None:
|
|||
|
||||
if __name__ == "__main__":
|
||||
plan, makespan = loese()
|
||||
print(f"Makespan: {makespan}")
|
||||
schranke = untere_schranke()
|
||||
print(f"Makespan: {makespan}, untere Schranke: {schranke}")
|
||||
for maschine, von, bis in stillstaende(plan):
|
||||
print(f" Stillstand: {MASCHINENNAMEN[maschine]} von {von} bis {bis}")
|
||||
|
||||
# Gegenprobe zu den Zahlen, die das Kapitel abdruckt.
|
||||
if schranke > makespan:
|
||||
raise SystemExit(f"Gantt: Die untere Schranke {schranke} liegt ueber "
|
||||
f"dem Makespan {makespan} - eine der beiden Zahlen "
|
||||
f"ist falsch.")
|
||||
if (schranke, makespan) != (10, 11):
|
||||
raise SystemExit(f"Gantt: Schranke {schranke} / Makespan {makespan}; "
|
||||
f"das Kapitel druckt 10 / 11 ab.")
|
||||
if not any(m == 1 and von == 4 for m, von, _ in stillstaende(plan)):
|
||||
raise SystemExit("Gantt: Die Luecke in der Dreherei bei Zeitpunkt 4 "
|
||||
"fehlt - der Text daneben bespricht genau sie.")
|
||||
|
||||
# Der Text daneben rechnet vor: Die Engpassmaschine ist von 0 bis zum
|
||||
# Makespan belegt, ihre Arbeit ist die Schranke, der Rest Stillstand.
|
||||
last = [0] * len(MASCHINENNAMEN)
|
||||
for auftrag in AUFTRAEGE:
|
||||
for maschine, dauer in auftrag:
|
||||
last[maschine] += dauer
|
||||
engpass = last.index(max(last))
|
||||
leerlauf = sum(bis - von for m, von, bis in stillstaende(plan)
|
||||
if m == engpass)
|
||||
if last[engpass] + leerlauf != makespan:
|
||||
raise SystemExit(
|
||||
f"Gantt: {MASCHINENNAMEN[engpass]} arbeitet {last[engpass]} und "
|
||||
f"steht {leerlauf} still, zusammen "
|
||||
f"{last[engpass] + leerlauf} statt {makespan}. Der Text daneben "
|
||||
f"rechnet genau diese Summe vor.")
|
||||
print(f" Engpass {MASCHINENNAMEN[engpass]}: {last[engpass]} Arbeit + "
|
||||
f"{leerlauf} Stillstand = {makespan}")
|
||||
|
||||
zeichne_statisch(plan, makespan)
|
||||
zeichne_interaktiv(plan, makespan)
|
||||
|
|
|
|||
214
bilder_04/erzeuge_or_architektur.py
Normal file
214
bilder_04/erzeuge_or_architektur.py
Normal file
|
|
@ -0,0 +1,214 @@
|
|||
#!/usr/bin/env python3
|
||||
|
||||
# erzeuge_or_architektur.py
|
||||
"""
|
||||
Erzeugt das Schichtenbild der OR-Plattform im Kapitel Praxisfallen:
|
||||
|
||||
bilder_04/kap15_or_plattform.svg
|
||||
|
||||
ERSETZT die bisherige, von Hand gezeichnete Fassung. Der Anlass war ein
|
||||
Widerspruch: Die Tabelle im Abschnitt nennt SECHS Schichten, das gezeichnete
|
||||
Bild daneben zeigte nur FUENF. Die sechste - "Betrieb" mit Monitoring,
|
||||
Audit-Trail und Alarmen - fehlte; "Audit-Trail" kam nur noch als Beschriftung
|
||||
eines Pfeils vor. Genau so entstehen Bilder, die dem Text daneben
|
||||
widersprechen, und genau deshalb bekommt dieses hier einen Generator.
|
||||
|
||||
Die Schichten werden NICHT in diesem Skript aufgezaehlt, sondern aus der
|
||||
Tabelle des Kapitels gelesen. Damit gibt es nur eine Quelle: Wer eine Zeile in
|
||||
50_Praxis.md ergaenzt, ergaenzt das Bild mit. Fehlt die Tabelle oder aendert
|
||||
sich ihr Aufbau, bricht der Generator ab, statt ein halbes Bild zu schreiben.
|
||||
|
||||
Dargestellt wird die Kette als das, was sie ist: Die ersten fuenf Schichten
|
||||
sind ein Datenweg von den Quellsystemen bis zur Ausfuehrung. "Betrieb" ist
|
||||
keine Station auf diesem Weg, sondern liegt neben allen - Monitoring und
|
||||
Audit-Trail betreffen jede Schicht. Als sechster Kasten in der Reihe waere sie
|
||||
falsch dargestellt; deshalb steht sie als Spange daneben. Dass sie beim
|
||||
Handzeichnen ganz verschwand, hat vermutlich denselben Grund.
|
||||
|
||||
Aufruf (aus dem Repository-Wurzelverzeichnis):
|
||||
python3 bilder_04/erzeuge_or_architektur.py
|
||||
|
||||
Benoetigt: matplotlib
|
||||
"""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import os
|
||||
import re
|
||||
import sys
|
||||
|
||||
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
|
||||
from stil_04 import FARBEN, speichere # noqa: E402
|
||||
import matplotlib.pyplot as plt # noqa: E402
|
||||
from matplotlib.patches import FancyArrowPatch, FancyBboxPatch # noqa: E402
|
||||
|
||||
BASIS = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
|
||||
KAPITEL = os.path.join(BASIS, "Operations_Research_mit_Python_Version_04",
|
||||
"50_Praxis.md")
|
||||
UEBERSCHRIFT = "## Architektur einer produktionsreifen OR-Plattform"
|
||||
|
||||
# Was zwischen den Schichten fliesst. Steht nicht in der Tabelle - die
|
||||
# beschreibt die Schichten, nicht die Uebergaenge. Fuenf Stationen in der
|
||||
# Kette ergeben VIER Pfeile; lies_schichten() rechnet das nach.
|
||||
UEBERGAENGE = [
|
||||
"Events · REST · Polling",
|
||||
"geprüfter Snapshot (JSON)",
|
||||
"Ergebnis + Kostenzerlegung",
|
||||
"Freigabe",
|
||||
]
|
||||
|
||||
# Die Schicht, die neben der Kette steht statt in ihr.
|
||||
QUERSCHNITT = "Betrieb"
|
||||
|
||||
|
||||
def saeubere(zelle: str) -> str:
|
||||
"""Markdown-Auszeichnung und Indexmarken aus einer Tabellenzelle nehmen."""
|
||||
zelle = re.sub(r"\{idx:[^}]*\}", "", zelle)
|
||||
zelle = zelle.replace("**", "").replace("`", "")
|
||||
return zelle.strip()
|
||||
|
||||
|
||||
def lies_schichten() -> list[tuple[str, str, str]]:
|
||||
"""Die Schichtentabelle des Kapitels als Liste (Schicht, Aufgabe, Werkzeuge).
|
||||
|
||||
Einzige Quelle fuer den Bildinhalt - siehe Modulkommentar."""
|
||||
with open(KAPITEL, encoding="utf-8") as datei:
|
||||
text = datei.read()
|
||||
if UEBERSCHRIFT not in text:
|
||||
raise SystemExit(f"OR-Architektur: Überschrift '{UEBERSCHRIFT}' nicht "
|
||||
f"in {os.path.basename(KAPITEL)} gefunden.")
|
||||
rest = text.split(UEBERSCHRIFT, 1)[1]
|
||||
|
||||
zeilen = []
|
||||
gefunden = False
|
||||
for zeile in rest.splitlines():
|
||||
if zeile.startswith("| Schicht |"):
|
||||
gefunden = True
|
||||
continue
|
||||
if not gefunden:
|
||||
continue
|
||||
if zeile.startswith("| ---"):
|
||||
continue
|
||||
if not zeile.startswith("|"):
|
||||
break
|
||||
felder = [saeubere(f) for f in zeile.strip().strip("|").split("|")]
|
||||
if len(felder) != 3:
|
||||
raise SystemExit(f"OR-Architektur: Tabellenzeile mit "
|
||||
f"{len(felder)} statt 3 Spalten: {zeile!r}")
|
||||
zeilen.append(tuple(felder))
|
||||
|
||||
if not zeilen:
|
||||
raise SystemExit("OR-Architektur: Die Schichtentabelle wurde nicht "
|
||||
"gefunden. Hat sich der Abschnitt geändert?")
|
||||
if zeilen[-1][0] != QUERSCHNITT:
|
||||
raise SystemExit(
|
||||
f"OR-Architektur: Die letzte Tabellenzeile heißt "
|
||||
f"'{zeilen[-1][0]}', erwartet wurde '{QUERSCHNITT}'. Diese "
|
||||
f"Schicht wird als Spange neben der Kette gezeichnet, nicht in "
|
||||
f"ihr - die Zuordnung muss stimmen.")
|
||||
# Die Kette sind alle Zeilen ausser der Querschnittsschicht; zwischen n
|
||||
# Stationen liegen n-1 Pfeile.
|
||||
if len(zeilen) - 2 != len(UEBERGAENGE):
|
||||
raise SystemExit(
|
||||
f"OR-Architektur: {len(zeilen) - 1} Stationen in der Kette "
|
||||
f"brauchen {len(zeilen) - 2} beschriftete Übergänge, es sind "
|
||||
f"aber {len(UEBERGAENGE)}. Beim Ergänzen einer Schicht gehört "
|
||||
f"ein Übergang dazu.")
|
||||
return zeilen
|
||||
|
||||
|
||||
def umbreche(text: str, breite: int) -> str:
|
||||
"""Einfacher Wortumbruch - textwrap.fill ohne den Import."""
|
||||
zeilen, aktuell = [], ""
|
||||
for wort in text.split():
|
||||
if aktuell and len(aktuell) + 1 + len(wort) > breite:
|
||||
zeilen.append(aktuell)
|
||||
aktuell = wort
|
||||
else:
|
||||
aktuell = f"{aktuell} {wort}".strip()
|
||||
if aktuell:
|
||||
zeilen.append(aktuell)
|
||||
return "\n".join(zeilen)
|
||||
|
||||
|
||||
def zeichne(schichten) -> None:
|
||||
kette = schichten[:-1]
|
||||
quer = schichten[-1]
|
||||
|
||||
figur, achse = plt.subplots(figsize=(8.8, 6.6))
|
||||
achse.set_axis_off()
|
||||
|
||||
kastenbreite, kastenhoehe = 6.4, 1.24
|
||||
abstand = 1.88
|
||||
links = 0.0
|
||||
|
||||
for stufe, (name, aufgabe, werkzeuge) in enumerate(kette):
|
||||
unten = -stufe * abstand
|
||||
achse.add_patch(FancyBboxPatch(
|
||||
(links, unten), kastenbreite, kastenhoehe,
|
||||
boxstyle="round,pad=0.02,rounding_size=0.10",
|
||||
facecolor=FARBEN["flaeche"], edgecolor=FARBEN["haupt"],
|
||||
linewidth=1.6, zorder=3))
|
||||
achse.text(links + 0.18, unten + kastenhoehe - 0.26, name,
|
||||
fontsize=10.5, color=FARBEN["haupt"], fontweight="bold",
|
||||
va="center", zorder=4)
|
||||
achse.text(links + 0.18, unten + kastenhoehe - 0.52,
|
||||
umbreche(aufgabe, 66), fontsize=8.5,
|
||||
color=FARBEN["text"], va="top", zorder=4)
|
||||
achse.text(links + 0.18, unten + 0.17,
|
||||
umbreche(werkzeuge, 70), fontsize=8,
|
||||
color=FARBEN["gedaempft"], va="bottom", style="italic",
|
||||
zorder=4)
|
||||
|
||||
if stufe < len(kette) - 1:
|
||||
pfeilspitze = unten - abstand + kastenhoehe
|
||||
achse.add_patch(FancyArrowPatch(
|
||||
(links + kastenbreite / 2, unten),
|
||||
(links + kastenbreite / 2, pfeilspitze),
|
||||
arrowstyle="-|>", mutation_scale=16,
|
||||
color=FARBEN["zweit"], linewidth=1.6, zorder=2))
|
||||
achse.text(links + kastenbreite / 2 + 0.16,
|
||||
(unten + pfeilspitze) / 2, UEBERGAENGE[stufe],
|
||||
fontsize=8, color=FARBEN["zweit"], va="center",
|
||||
zorder=4)
|
||||
|
||||
# Die Querschnittsschicht als Spange neben der ganzen Kette.
|
||||
unten_ganz = -(len(kette) - 1) * abstand
|
||||
spange_links = links + kastenbreite + 0.45
|
||||
achse.add_patch(FancyBboxPatch(
|
||||
(spange_links, unten_ganz), 0.92,
|
||||
(len(kette) - 1) * abstand + kastenhoehe,
|
||||
boxstyle="round,pad=0.02,rounding_size=0.10",
|
||||
facecolor=FARBEN["warnung"], alpha=0.13,
|
||||
edgecolor=FARBEN["warnung"], linewidth=1.6, zorder=3))
|
||||
achse.text(spange_links + 0.46,
|
||||
unten_ganz + ((len(kette) - 1) * abstand + kastenhoehe) / 2,
|
||||
f"{quer[0]} — {quer[1]}\n{quer[2]}",
|
||||
fontsize=9, color=FARBEN["warnung"], fontweight="bold",
|
||||
ha="center", va="center", rotation=90, zorder=4)
|
||||
|
||||
achse.text(links, kastenhoehe + 0.42,
|
||||
"Der Optimierer ist kein Skript, sondern ein Dienst",
|
||||
fontsize=11.5, color=FARBEN["text"], fontweight="bold")
|
||||
achse.text(links, -(len(kette) - 1) * abstand - 0.42,
|
||||
f"„{quer[0]}“ ist keine Station auf dem Weg — Monitoring und "
|
||||
f"Audit-Trail betreffen jede Schicht.\nDeshalb steht sie "
|
||||
f"neben der Kette, nicht in ihr.",
|
||||
fontsize=8.5, color=FARBEN["gedaempft"], va="top")
|
||||
|
||||
achse.set_xlim(-0.3, spange_links + 1.5)
|
||||
achse.set_ylim(-(len(kette) - 1) * abstand - 1.3, kastenhoehe + 0.85)
|
||||
achse.set_aspect("equal")
|
||||
|
||||
speichere(figur, "kap15_or_plattform")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
schichten = lies_schichten()
|
||||
for name, aufgabe, werkzeuge in schichten:
|
||||
markierung = " (Spange)" if name == QUERSCHNITT else ""
|
||||
print(f" {name:<22}{markierung}")
|
||||
print(f" {len(schichten)} Schichten aus der Tabelle in "
|
||||
f"{os.path.basename(KAPITEL)} gelesen — "
|
||||
f"{len(schichten) - 1} in der Kette, 1 quer.")
|
||||
zeichne(schichten)
|
||||
|
|
@ -107,6 +107,72 @@ def untere_huelle(punkte):
|
|||
return huelle
|
||||
|
||||
|
||||
def tiefste_delle(front, huelle):
|
||||
"""Der unerreichbare Punkt mit dem groessten Abstand zur Huelle.
|
||||
|
||||
Liefert (Punkt, linke Huellenecke, rechte Huellenecke, Abstand). Genau an
|
||||
diesem Punkt ist die Luecke am besten zu sehen, und genau dort wird sie
|
||||
beschriftet - gesucht wird sie, nicht ausgewaehlt.
|
||||
|
||||
Gemessen wird SENKRECHT zur Huellenkante, nicht senkrecht nach oben. Der
|
||||
senkrechte Abstand ist der Abstand eines Punktes zu einer Geraden; der
|
||||
Abstand in y-Richtung haengt dagegen davon ab, welche Zielgroesse man auf
|
||||
die Hochachse legt - bei vertauschten Achsen faende er eine andere Delle."""
|
||||
unerreichbar = [p for p in front if p not in huelle]
|
||||
if not unerreichbar:
|
||||
return None
|
||||
bestes = None
|
||||
for punkt in unerreichbar:
|
||||
for links, rechts in zip(huelle, huelle[1:]):
|
||||
if not links[0] <= punkt[0] <= rechts[0]:
|
||||
continue
|
||||
if rechts[0] == links[0]:
|
||||
continue
|
||||
steigung = (rechts[1] - links[1]) / (rechts[0] - links[0])
|
||||
auf_huelle = links[1] + steigung * (punkt[0] - links[0])
|
||||
abstand = (punkt[1] - auf_huelle) / np.hypot(1.0, steigung)
|
||||
if bestes is None or abstand > bestes[3]:
|
||||
bestes = (punkt, links, rechts, abstand)
|
||||
return bestes
|
||||
|
||||
|
||||
def zeichne_luecke(achse, front, huelle) -> None:
|
||||
"""Warum die gewichtete Summe die Delle nicht erreicht - und was hilft.
|
||||
|
||||
Gezeichnet werden zwei Geraden derselben Steigung: die Tangente an die
|
||||
Huelle und die parallele Gerade durch den unerreichbaren Punkt. Eine
|
||||
gewichtete Summe minimiert genau diese Steigung; sie waehlt also immer die
|
||||
untere der beiden Geraden und damit nie den Punkt darueber. Der
|
||||
waagerechte Strich zeigt die Alternative: eine CO2-Schranke, unter der
|
||||
dieser Punkt der billigste ist - so ist die Front oben ueberhaupt
|
||||
entstanden."""
|
||||
delle = tiefste_delle(front, huelle)
|
||||
if delle is None:
|
||||
return
|
||||
punkt, links, rechts, _abstand = delle
|
||||
steigung = ((rechts[1] - links[1]) / (rechts[0] - links[0])
|
||||
if rechts[0] != links[0] else 0.0)
|
||||
|
||||
x = np.array(achse.get_xlim())
|
||||
achse.plot(x, links[1] + steigung * (x - links[0]), "-", color=INDIGO,
|
||||
linewidth=1.1, alpha=0.75, zorder=1)
|
||||
achse.plot(x, punkt[1] + steigung * (x - punkt[0]), ":", color=AMBER,
|
||||
linewidth=1.6, zorder=2)
|
||||
achse.annotate("dieselbe Steigung, eine Stufe höher —\n"
|
||||
"jede gewichtete Summe nimmt die untere",
|
||||
xy=(punkt[0], punkt[1]), xytext=(12, 26),
|
||||
textcoords="offset points", fontsize=7.5, color=AMBER,
|
||||
zorder=6)
|
||||
|
||||
achse.axhline(punkt[1], color="#059669", linestyle="--", linewidth=1.2,
|
||||
alpha=0.9, zorder=1)
|
||||
achse.annotate(f"ε-Constraint: CO₂ ≤ {punkt[1]:,} kg".replace(",", ".")
|
||||
+ "\nfindet genau diesen Punkt",
|
||||
xy=(x[0], punkt[1]), xytext=(6, -22),
|
||||
textcoords="offset points", fontsize=7.5, color="#059669",
|
||||
zorder=6)
|
||||
|
||||
|
||||
# --- Statische Fassung ------------------------------------------------------
|
||||
|
||||
def zeichne_statisch(front, huelle) -> None:
|
||||
|
|
@ -132,6 +198,9 @@ def zeichne_statisch(front, huelle) -> None:
|
|||
achse.set_xlabel("Transportkosten (EUR)")
|
||||
achse.set_ylabel("CO₂-Ausstoß (kg)")
|
||||
achse.set_title("Pareto-Front: Kosten gegen CO₂", fontsize=11)
|
||||
# Erst jetzt, damit die Geraden die endgueltigen Achsengrenzen benutzen.
|
||||
achse.autoscale(enable=False)
|
||||
zeichne_luecke(achse, front, huelle)
|
||||
achse.grid(linestyle=":", alpha=0.5)
|
||||
for rand in ("top", "right"):
|
||||
achse.spines[rand].set_visible(False)
|
||||
|
|
@ -203,5 +272,49 @@ if __name__ == "__main__":
|
|||
huelle = untere_huelle(front)
|
||||
print(f"Pareto-Punkte: {len(front)}, davon gestuetzt: {len(huelle)}, "
|
||||
f"unerreichbar: {len(front) - len(huelle)}")
|
||||
|
||||
# Die beiden Behauptungen des Luecken-Bildes nachrechnen, statt sie
|
||||
# hinzuschreiben.
|
||||
delle = tiefste_delle(front, huelle)
|
||||
if delle is not None:
|
||||
punkt, links, rechts, abstand = delle
|
||||
steigung = (rechts[1] - links[1]) / (rechts[0] - links[0])
|
||||
|
||||
def auf_tangente(x):
|
||||
return links[1] + steigung * (x - links[0])
|
||||
|
||||
# Eine Stuetzgerade der UNTEREN Huelle liegt unter allen Punkten und
|
||||
# beruehrt die beiden Ecken der Kante. Beides zusammen macht sie zur
|
||||
# Tangente; nur eines davon reicht nicht.
|
||||
for ecke in huelle:
|
||||
if ecke[1] < auf_tangente(ecke[0]) - 1e-6:
|
||||
raise SystemExit(
|
||||
f"Pareto-Luecke: Die Huellenecke {ecke} liegt UNTER der "
|
||||
f"gezeichneten Geraden - dann stuetzt sie die Huelle "
|
||||
f"nicht.")
|
||||
for ecke in (links, rechts):
|
||||
if abs(ecke[1] - auf_tangente(ecke[0])) > 1e-6:
|
||||
raise SystemExit(
|
||||
f"Pareto-Luecke: Die Gerade beruehrt die Huellenecke "
|
||||
f"{ecke} nicht - dann ist sie keine Tangente an diese "
|
||||
f"Kante.")
|
||||
if punkt[1] <= auf_tangente(punkt[0]) + 1e-6:
|
||||
raise SystemExit(
|
||||
f"Pareto-Luecke: {punkt} liegt nicht ueber der Tangente - "
|
||||
f"dann waere der Punkt von einer gewichteten Summe erreichbar.")
|
||||
gefunden = plane(KOSTEN, co2_grenze=punkt[1])
|
||||
if gefunden != punkt:
|
||||
raise SystemExit(
|
||||
f"Pareto-Luecke: Die Schranke CO2 <= {punkt[1]} liefert "
|
||||
f"{gefunden}, nicht {punkt}. Die Bildunterschrift zum "
|
||||
f"ε-Constraint waere falsch.")
|
||||
# Die Zahl getrennt formatieren: Ein .replace(",", ".") auf den ganzen
|
||||
# f-String erwischt auch das Komma im Tupel.
|
||||
abstand_text = f"{abstand:.1f}".replace(".", ",")
|
||||
print(f" tiefste Delle: {punkt}, Abstand zur Hülle "
|
||||
f"{abstand_text} (senkrecht)")
|
||||
print(f" Tangente stützt die Hülle, der Punkt liegt darüber")
|
||||
print(f" ε-Constraint CO₂ ≤ {punkt[1]} liefert genau diesen Punkt")
|
||||
|
||||
zeichne_statisch(front, huelle)
|
||||
zeichne_interaktiv(front, huelle)
|
||||
|
|
|
|||
|
|
@ -50,6 +50,7 @@ BASIS = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
|
|||
BILDER = os.path.join(BASIS, "bilder_04")
|
||||
|
||||
INDIGO, CYAN, AMBER, GRUEN = "#4338ca", "#0891b2", "#b45309", "#059669"
|
||||
VIOLETT = "#7c3aed" # Niveaulinien und Zielfunktionsgradient
|
||||
|
||||
# Modell wortgleich zu Visualisierung_Loesungsraum.py
|
||||
C = np.array([150.0, 250.0])
|
||||
|
|
@ -200,12 +201,46 @@ def zeichne_polyeder(ecken: np.ndarray, bestes: np.ndarray) -> None:
|
|||
schreibe(figur, "kap_fundament_polyeder")
|
||||
|
||||
|
||||
def zeichne_niveaulinien(achse, besucht: list[np.ndarray]) -> None:
|
||||
"""Eine Niveaulinie der Zielfunktion durch JEDE besuchte Ecke.
|
||||
|
||||
Die Niveaus werden aus dem Weg gerechnet, nicht gewaehlt: Jede Ecke des
|
||||
Simplexwegs liegt damit auf genau einer Linie, und der Abstand der Linien
|
||||
ist der Fortschritt eines Schritts. Damit zeigt das Bild nicht nur, WELCHEN
|
||||
Weg der Simplex nimmt, sondern warum - er schiebt die Linie so weit nach
|
||||
aussen, wie das Polyeder es zulaesst."""
|
||||
x = np.linspace(-0.9, 11.4, 100)
|
||||
for punkt in besucht:
|
||||
niveau = float(C @ punkt)
|
||||
achse.plot(x, (niveau - C[0] * x) / C[1], linestyle=":",
|
||||
color=VIOLETT, linewidth=1.3, alpha=0.9, zorder=2)
|
||||
# Beschriftung dort, wo die Linie im Bild noch Platz hat. Die
|
||||
# Nulllinie laeuft nahe am unteren Rand - ihre Beschriftung gehoert
|
||||
# darueber, sonst wird sie abgeschnitten.
|
||||
unten = niveau < 500
|
||||
x_text = 1.7 if unten else 10.1
|
||||
y_linie = (niveau - C[0] * x_text) / C[1]
|
||||
achse.text(x_text, y_linie + (0.16 if unten else -0.28),
|
||||
f"Z = {euro(niveau)}", fontsize=7.5, color=VIOLETT,
|
||||
ha="center", va="bottom" if unten else "top", zorder=6)
|
||||
|
||||
# Der Gradient: die Richtung, in die sich die Linien verschieben lassen.
|
||||
start = np.array([2.6, 2.2])
|
||||
ende = start + C / np.linalg.norm(C) * 2.6
|
||||
achse.annotate("", xy=tuple(ende), xytext=tuple(start),
|
||||
arrowprops=dict(arrowstyle="-|>", color=VIOLETT,
|
||||
linewidth=2.2), zorder=6)
|
||||
achse.text(ende[0] + 0.15, ende[1], "$\\mathbf{c} = (150; 250)$",
|
||||
fontsize=8.5, color=VIOLETT, va="center", zorder=6)
|
||||
|
||||
|
||||
def zeichne_simplexweg(ecken: np.ndarray, bestes: np.ndarray) -> None:
|
||||
"""LP-Kapitel: derselbe Bereich, aber der Weg der Handrechnung."""
|
||||
figur, achse = plt.subplots(figsize=(7.2, 5.2))
|
||||
zeichne_rahmen(achse, ecken)
|
||||
|
||||
besucht = [np.array(p) for p in SIMPLEXWEG]
|
||||
zeichne_niveaulinien(achse, besucht)
|
||||
unbesucht = [e for e in ecken
|
||||
if not any(np.allclose(e, p) for p in besucht)]
|
||||
for ecke in unbesucht:
|
||||
|
|
|
|||
168
bilder_04/erzeuge_propagation_ablauf.py
Normal file
168
bilder_04/erzeuge_propagation_ablauf.py
Normal file
|
|
@ -0,0 +1,168 @@
|
|||
#!/usr/bin/env python3
|
||||
|
||||
# erzeuge_propagation_ablauf.py
|
||||
"""
|
||||
Erzeugt die Ablaufgrafik zur Propagation im Kapitel CP-SAT:
|
||||
|
||||
bilder_04/kap_cpsat_propagation_ablauf.svg
|
||||
|
||||
ERGAENZT das vorhandene kap06_cp_sat_propagation.svg, ersetzt es nicht: Jenes
|
||||
zeigt schematisch das Zusammenspiel von Propagation und Konfliktlernen. Dieses
|
||||
zeigt die Propagation selbst - Runde fuer Runde, an Zahlen.
|
||||
|
||||
Gezeigt wird GENAU die Handrechnung des Kapitels:
|
||||
|
||||
x1 in {1,2,3,4}, x2 in {2,3,4,5}, Bedingung x1 + x2 = 7
|
||||
|
||||
Ausgangslage 4 x 4 = 16 Kombinationen
|
||||
nach Runde 1 x1 -> {2,3,4} 12 Kombinationen
|
||||
nach Runde 2 x2 -> {3,4,5} 9 Kombinationen
|
||||
|
||||
Bewusst nicht das Beispiel aus der Auftragsbeschreibung ([0,10] mit x1+x2<=6):
|
||||
Die Handrechnung steht unmittelbar neben dem Bild. Ein Bild mit anderen Zahlen
|
||||
als der Text daneben zwingt den Leser, beide zu vergleichen, statt eines zu
|
||||
verstehen.
|
||||
|
||||
Die Domains werden PROPAGIERT, nicht eingetragen: revidiere() streicht jeden
|
||||
Wert, fuer den es im Partner keinen Partner gibt - das ist Kantenkonsistenz,
|
||||
und zwar in derselben Reihenfolge wie in der Handrechnung. Der Selbsttest
|
||||
vergleicht das Ergebnis mit den drei im Kapitel abgedruckten Zahlen.
|
||||
|
||||
Aufruf (aus dem Repository-Wurzelverzeichnis):
|
||||
python3 bilder_04/erzeuge_propagation_ablauf.py
|
||||
|
||||
Benoetigt: matplotlib
|
||||
"""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import os
|
||||
import sys
|
||||
|
||||
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
|
||||
from stil_04 import FARBEN, speichere # noqa: E402
|
||||
import matplotlib.pyplot as plt # noqa: E402
|
||||
from matplotlib.patches import Rectangle # noqa: E402
|
||||
|
||||
# Die Aufgabe der Handrechnung.
|
||||
START_X1 = {1, 2, 3, 4}
|
||||
START_X2 = {2, 3, 4, 5}
|
||||
SUMME = 7
|
||||
ANZEIGE = range(1, 7) # 1 bis 6 - alle je vorkommenden Werte
|
||||
|
||||
# Was das Kapitel abdruckt; nur zum Gegenpruefen.
|
||||
KOMBINATIONEN_IM_BUCH = [16, 12, 9]
|
||||
ENDDOMAINS_IM_BUCH = ({2, 3, 4}, {3, 4, 5})
|
||||
|
||||
|
||||
def revidiere(eigene: set[int], partner: set[int]) -> set[int]:
|
||||
"""Streicht jeden Wert, zu dem es im Partner keinen Partner gibt."""
|
||||
return {v for v in eigene if (SUMME - v) in partner}
|
||||
|
||||
|
||||
def propagiere():
|
||||
"""Die zwei Runden der Handrechnung, jede als eigener Zustand."""
|
||||
d1, d2 = set(START_X1), set(START_X2)
|
||||
verlauf = [("Ausgangslage", set(d1), set(d2), None)]
|
||||
|
||||
d1 = revidiere(d1, d2)
|
||||
verlauf.append((f"Runde 1 — von $x_1$ aus", set(d1), set(d2), 0))
|
||||
|
||||
d2 = revidiere(d2, d1)
|
||||
verlauf.append((f"Runde 2 — von $x_2$ aus", set(d1), set(d2), 1))
|
||||
return verlauf
|
||||
|
||||
|
||||
def zeichne(verlauf) -> None:
|
||||
figur, achse = plt.subplots(figsize=(8.6, 4.6))
|
||||
achse.set_axis_off()
|
||||
|
||||
zellbreite, zellhoehe = 1.0, 0.62
|
||||
zeilenabstand = 1.05
|
||||
blockabstand = 2.6
|
||||
|
||||
for stufe, (titel, d1, d2, geaendert) in enumerate(verlauf):
|
||||
oben = -stufe * blockabstand
|
||||
achse.text(-0.4, oben + 0.75, titel, fontsize=10,
|
||||
color=FARBEN["text"], fontweight="bold", ha="left",
|
||||
va="center")
|
||||
|
||||
for reihe, (name, domain, vorher) in enumerate((
|
||||
("$x_1$", d1, verlauf[stufe - 1][1] if stufe else d1),
|
||||
("$x_2$", d2, verlauf[stufe - 1][2] if stufe else d2))):
|
||||
y = oben - reihe * zeilenabstand
|
||||
achse.text(-0.4, y + zellhoehe / 2, name, fontsize=11,
|
||||
color=FARBEN["text"], ha="left", va="center")
|
||||
for spalte, wert in enumerate(ANZEIGE):
|
||||
x = spalte * zellbreite
|
||||
gerade_gestrichen = wert in vorher and wert not in domain
|
||||
if wert in domain:
|
||||
flaeche, rand, schrift = (FARBEN["gut"], FARBEN["gut"],
|
||||
"white")
|
||||
alpha = 0.85
|
||||
elif gerade_gestrichen:
|
||||
flaeche, rand, schrift = (FARBEN["fehler"],
|
||||
FARBEN["fehler"], "white")
|
||||
alpha = 0.75
|
||||
else:
|
||||
flaeche, rand, schrift = (FARBEN["flaeche"],
|
||||
FARBEN["linie"],
|
||||
FARBEN["linie"])
|
||||
alpha = 1.0
|
||||
achse.add_patch(Rectangle((x, y), zellbreite * 0.9, zellhoehe,
|
||||
facecolor=flaeche, edgecolor=rand,
|
||||
alpha=alpha, linewidth=1.0))
|
||||
achse.text(x + zellbreite * 0.45, y + zellhoehe / 2, str(wert),
|
||||
fontsize=10, color=schrift, ha="center",
|
||||
va="center", fontweight="bold")
|
||||
if gerade_gestrichen:
|
||||
achse.plot([x + 0.1, x + zellbreite * 0.8],
|
||||
[y + zellhoehe - 0.08, y + 0.08], "-",
|
||||
color="white", linewidth=1.6)
|
||||
|
||||
anzahl = len(d1) * len(d2)
|
||||
achse.text(len(ANZEIGE) * zellbreite + 0.35, oben - 0.22,
|
||||
f"{len(d1)} × {len(d2)} = {anzahl}\nKombinationen",
|
||||
fontsize=9.5, color=FARBEN["text"], va="center")
|
||||
if geaendert is not None:
|
||||
gestrichen = sorted(verlauf[stufe - 1][1 + geaendert]
|
||||
- (d1, d2)[geaendert])
|
||||
achse.text(len(ANZEIGE) * zellbreite + 0.35, oben - 1.15,
|
||||
"gestrichen: "
|
||||
+ ", ".join(str(w) for w in gestrichen),
|
||||
fontsize=9, color=FARBEN["fehler"], va="center")
|
||||
|
||||
achse.text(-0.4, -2 * blockabstand - 1.65,
|
||||
f"Bedingung $x_1 + x_2 = {SUMME}$ — kein einziger Wert wurde "
|
||||
f"ausprobiert. Gestrichen wird,\nwofür es im Partner keinen "
|
||||
f"Partner gibt.",
|
||||
fontsize=9.5, color=FARBEN["text"], va="top")
|
||||
|
||||
achse.set_xlim(-0.6, len(ANZEIGE) * zellbreite + 2.6)
|
||||
achse.set_ylim(-2 * blockabstand - 2.5, 1.2)
|
||||
achse.set_aspect("equal")
|
||||
|
||||
speichere(figur, "kap_cpsat_propagation_ablauf")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
verlauf = propagiere()
|
||||
for titel, d1, d2, _ in verlauf:
|
||||
reintitel = titel.replace("$", "").replace("\\", "")
|
||||
print(f" {reintitel:<24} x1 = {sorted(d1)} x2 = {sorted(d2)} "
|
||||
f"-> {len(d1) * len(d2)} Kombinationen")
|
||||
|
||||
anzahlen = [len(d1) * len(d2) for _, d1, d2, _ in verlauf]
|
||||
if anzahlen != KOMBINATIONEN_IM_BUCH:
|
||||
raise SystemExit(
|
||||
f"Propagations-Ablauf: Die Propagation ergibt {anzahlen} "
|
||||
f"Kombinationen, das Kapitel druckt {KOMBINATIONEN_IM_BUCH} ab.")
|
||||
if (verlauf[-1][1], verlauf[-1][2]) != ENDDOMAINS_IM_BUCH:
|
||||
raise SystemExit(
|
||||
f"Propagations-Ablauf: Enddomains {sorted(verlauf[-1][1])} / "
|
||||
f"{sorted(verlauf[-1][2])} weichen von der Handrechnung "
|
||||
f"{sorted(ENDDOMAINS_IM_BUCH[0])} / "
|
||||
f"{sorted(ENDDOMAINS_IM_BUCH[1])} ab.")
|
||||
print(" stimmt mit der Handrechnung überein: 16 → 12 → 9")
|
||||
|
||||
zeichne(verlauf)
|
||||
169
bilder_04/erzeuge_var_vs_cvar.py
Normal file
169
bilder_04/erzeuge_var_vs_cvar.py
Normal file
|
|
@ -0,0 +1,169 @@
|
|||
#!/usr/bin/env python3
|
||||
|
||||
# erzeuge_var_vs_cvar.py
|
||||
"""
|
||||
Erzeugt das Subadditivitaets-Bild zum Kapitel CVaR:
|
||||
|
||||
bilder_04/kap_cvar_subadditivitaet.svg
|
||||
|
||||
ERGAENZT kap_cvar_var_vergleich.svg, ersetzt es nicht: Jenes zeigt, WAS die
|
||||
beiden Kennzahlen sind - eine Schwelle und ein Mittelwert. Dieses zeigt, warum
|
||||
der Unterschied zaehlt.
|
||||
|
||||
Gerechnet wird das Gegenbeispiel aus Teil 2 von VaR_CVaR_Demo.py: zwei
|
||||
unabhaengige Anleihen, je 100 EUR Nominal, Ausfallwahrscheinlichkeit 4 %,
|
||||
sonst 2 EUR Kupon. Einzeln meldet der VaR fuer jede Anleihe einen GEWINN von
|
||||
2 EUR - denn 4 % liegen unter der 5-%-Schwelle, der Ausfall faellt also aus der
|
||||
Betrachtung heraus. Im Portfolio steigt die Wahrscheinlichkeit mindestens eines
|
||||
Ausfalls auf 7,8 % und damit UEBER die Schwelle; der VaR springt auf 98 EUR.
|
||||
|
||||
Das ist die Aussage des Bildes: Der VaR behauptet, Streuung habe das Risiko um
|
||||
102 EUR ERHOEHT. Der CVaR, der den ganzen Rand mittelt statt ihn abzuschneiden,
|
||||
kommt auf 101,33 EUR gegenueber 159,24 EUR - Streuung senkt das Risiko, wie es
|
||||
sein muss.
|
||||
|
||||
Die Ziehungsreihenfolge des Buchprogramms wird nachgespielt: default_rng(2026),
|
||||
und Teil 1 des Programms zieht nichts, also sind die beiden Anleihen die ersten
|
||||
Ziehungen. Deshalb stehen im Bild genau die Zahlen der abgedruckten Tabelle;
|
||||
pruefe_gegen_buch() bricht ab, wenn nicht.
|
||||
|
||||
Aufruf (aus dem Repository-Wurzelverzeichnis):
|
||||
python3 bilder_04/erzeuge_var_vs_cvar.py
|
||||
|
||||
Benoetigt: numpy, matplotlib
|
||||
"""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import os
|
||||
import sys
|
||||
|
||||
import numpy as np
|
||||
|
||||
HIER = os.path.dirname(os.path.abspath(__file__))
|
||||
BASIS = os.path.dirname(HIER)
|
||||
sys.path.insert(0, HIER)
|
||||
sys.path.insert(0, os.path.join(
|
||||
BASIS, "Operations_Research_mit_Python_Version_04_Programme"))
|
||||
|
||||
from stil_04 import FARBEN, speichere # noqa: E402
|
||||
import matplotlib.pyplot as plt # noqa: E402
|
||||
from VaR_CVaR_Demo import cvar_rockafellar, var_quantil # noqa: E402
|
||||
|
||||
# Instanz wortgleich zu Teil 2 von VaR_CVaR_Demo.py.
|
||||
SAAT = 2026
|
||||
ZIEHUNGEN = 2_000_000
|
||||
AUSFALLWAHRSCHEINLICHKEIT = 0.04
|
||||
VERLUST_BEI_AUSFALL = 100.0
|
||||
KUPON = 2.0
|
||||
|
||||
# Die im Kapitel abgedruckte Tabelle; nur zum Gegenpruefen.
|
||||
TABELLE_IM_BUCH = {
|
||||
"Anleihe A allein": (-2.00, 79.59),
|
||||
"Anleihe B allein": (-2.00, 79.66),
|
||||
"Portfolio A+B": (98.00, 101.33),
|
||||
"Summe der Einzelwerte": (-4.00, 159.24),
|
||||
}
|
||||
|
||||
|
||||
def rechne() -> dict[str, tuple[float, float]]:
|
||||
"""Dieselben Ziehungen wie im Buchprogramm - siehe Modulkommentar."""
|
||||
rng = np.random.default_rng(SAAT)
|
||||
verlust_a = np.where(rng.random(ZIEHUNGEN) < AUSFALLWAHRSCHEINLICHKEIT,
|
||||
VERLUST_BEI_AUSFALL, -KUPON)
|
||||
verlust_b = np.where(rng.random(ZIEHUNGEN) < AUSFALLWAHRSCHEINLICHKEIT,
|
||||
VERLUST_BEI_AUSFALL, -KUPON)
|
||||
|
||||
werte = {}
|
||||
for name, v in (("Anleihe A allein", verlust_a),
|
||||
("Anleihe B allein", verlust_b),
|
||||
("Portfolio A+B", verlust_a + verlust_b)):
|
||||
werte[name] = (var_quantil(v), cvar_rockafellar(v))
|
||||
werte["Summe der Einzelwerte"] = tuple(
|
||||
werte["Anleihe A allein"][i] + werte["Anleihe B allein"][i]
|
||||
for i in (0, 1))
|
||||
return werte
|
||||
|
||||
|
||||
def pruefe_gegen_buch(werte) -> None:
|
||||
"""Die Tafel muss dieselben Zahlen zeigen wie die Tabelle im Kapitel."""
|
||||
for name, (soll_var, soll_cvar) in TABELLE_IM_BUCH.items():
|
||||
ist_var, ist_cvar = werte[name]
|
||||
if abs(ist_var - soll_var) > 0.005 or abs(ist_cvar - soll_cvar) > 0.005:
|
||||
raise SystemExit(
|
||||
f"Subadditivitaets-Bild: '{name}' ergibt VaR {ist_var:.2f} / "
|
||||
f"CVaR {ist_cvar:.2f}, das Kapitel druckt {soll_var:.2f} / "
|
||||
f"{soll_cvar:.2f} ab.")
|
||||
# Die Aussage selbst - nicht nur die Zahlen.
|
||||
if not werte["Portfolio A+B"][0] > werte["Summe der Einzelwerte"][0]:
|
||||
raise SystemExit("Subadditivitaets-Bild: Der VaR verletzt die "
|
||||
"Subadditivitaet hier nicht - dann zeigt die Tafel "
|
||||
"das Gegenteil ihrer Überschrift.")
|
||||
if not werte["Portfolio A+B"][1] <= werte["Summe der Einzelwerte"][1]:
|
||||
raise SystemExit("Subadditivitaets-Bild: Der CVaR ist hier nicht "
|
||||
"subadditiv. Das widerspricht dem Kapiteltext.")
|
||||
|
||||
|
||||
def euro(wert: float) -> str:
|
||||
return f"{wert:,.2f} €".replace(",", "@").replace(".", ",").replace("@", ".")
|
||||
|
||||
|
||||
def zeichne(werte) -> None:
|
||||
figur, achsen = plt.subplots(1, 2, figsize=(10.2, 4.4))
|
||||
namen = ["Anleihe A allein", "Anleihe B allein",
|
||||
"Summe der Einzelwerte", "Portfolio A+B"]
|
||||
y = np.arange(len(namen))[::-1]
|
||||
|
||||
for achse, index, titel, urteil, farbe in (
|
||||
(achsen[0], 0, "VaR 95 %",
|
||||
"Portfolio teurer als die Summe —\nSubadditivität verletzt",
|
||||
FARBEN["fehler"]),
|
||||
(achsen[1], 1, "CVaR 95 %",
|
||||
"Portfolio billiger als die Summe —\nsubadditiv, also kohärent",
|
||||
FARBEN["gut"])):
|
||||
wert = [werte[n][index] for n in namen]
|
||||
farben = [FARBEN["gedaempft"], FARBEN["gedaempft"],
|
||||
FARBEN["zweit"], farbe]
|
||||
achse.barh(y, wert, height=0.6, color=farben, zorder=3)
|
||||
# Balken muessen bei null beginnen, sonst luegt ihre Laenge. Der Rand
|
||||
# links richtet sich nach der Spannweite, damit die Zahl an einem
|
||||
# negativen Balken nicht an der Achsenbeschriftung klebt.
|
||||
spanne = max(wert) - min(0.0, min(wert))
|
||||
achse.set_xlim(min(0.0, min(wert)) - 0.20 * spanne, max(wert) * 1.22)
|
||||
for stelle, w in enumerate(wert):
|
||||
abstand = 0.015 * spanne
|
||||
achse.text(w + (abstand if w >= 0 else -abstand), y[stelle],
|
||||
euro(w), va="center", ha="left" if w >= 0 else "right",
|
||||
fontsize=9, color=FARBEN["text"], zorder=4)
|
||||
achse.axvline(0, color=FARBEN["text"], linewidth=1.0, zorder=2)
|
||||
achse.set_yticks(y)
|
||||
achse.set_yticklabels(namen if index == 0 else [], fontsize=9,
|
||||
color=FARBEN["text"])
|
||||
achse.set_title(titel, fontsize=11, color=FARBEN["text"], pad=34)
|
||||
achse.set_xlabel("Verlust in € (negativ = Gewinn)")
|
||||
achse.grid(axis="y", visible=False)
|
||||
# Das Urteil ueber der Achse, nicht in ihr - sonst stoesst es an die
|
||||
# Zahl am obersten Balken.
|
||||
achse.text(0.5, 1.008, urteil, transform=achse.transAxes,
|
||||
fontsize=8.5, color=farbe, ha="center", va="bottom",
|
||||
fontweight="bold")
|
||||
|
||||
figur.suptitle("Zwei unabhängige Anleihen, je 4 % Ausfallrisiko — "
|
||||
"und zwei Urteile über dieselbe Diversifikation",
|
||||
fontsize=11, color=FARBEN["text"], y=1.02)
|
||||
speichere(figur, "kap_cvar_subadditivitaet")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
werte = rechne()
|
||||
print(f" {'':<24} {'VaR 95%':>10} {'CVaR 95%':>10}")
|
||||
for name in ("Anleihe A allein", "Anleihe B allein", "Portfolio A+B",
|
||||
"Summe der Einzelwerte"):
|
||||
print(f" {name:<24} {werte[name][0]:>10.2f} {werte[name][1]:>10.2f}")
|
||||
pruefe_gegen_buch(werte)
|
||||
print(" stimmt mit der abgedruckten Tabelle überein")
|
||||
print(f" VaR: Portfolio {werte['Portfolio A+B'][0]:.2f} > Summe "
|
||||
f"{werte['Summe der Einzelwerte'][0]:.2f} — verletzt")
|
||||
print(f" CVaR: Portfolio {werte['Portfolio A+B'][1]:.2f} ≤ Summe "
|
||||
f"{werte['Summe der Einzelwerte'][1]:.2f} — kohärent")
|
||||
zeichne(werte)
|
||||
71417
bilder_04/kap02_konvexitaet_landschaft.svg
Normal file
71417
bilder_04/kap02_konvexitaet_landschaft.svg
Normal file
File diff suppressed because it is too large
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6157
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After Width: | Height: | Size: 160 KiB |
Loading…
Reference in a new issue