to/Tema 1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reprezentarea grafică a diferitelor tipuri de programe liniare"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un program liniar poate fi:\n",
    "\n",
    "- Incompatibil (fără soluție)\n",
    "- Compatibil:\n",
    "  - Cu o soluție optimă (de valoare finită)\n",
    "  - Cu o infinitate de soluții optime (de aceeași valoare)\n",
    "  - Cu optim infinit (nemărginită)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voi folosi `numpy` și `matplotlib` pentru a reprezenta grafic sistemele."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sistem incompatibil\n",
    "\n",
    "Luăm ca exemplu următorul program liniar:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\max x + y \\\\\n",
    "x + 2y \\geq 3 \\\\\n",
    "x - 2y \\geq 2 \\\\\n",
    "y \\leq -2\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Pentru a face afișarea, scriu expresiile de mai sus cu $y$ în funcție de $x$:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "y = \\frac{3 - x}{2} \\\\\n",
    "y = \\frac{x - 2}{2} \\\\\n",
    "y = -2\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-5, 5, 200)\n",
    "\n",
    "plt.figure(dpi=100)\n",
    "\n",
    "plt.xlim(-3, 5)\n",
    "plt.ylim(-4, 4)\n",
    "\n",
    "# x + 2y >= 3\n",
    "y = (3 - x)/2\n",
    "plt.plot(x, y)\n",
    "plt.fill_between(x, y, y + 10, alpha=0.2)\n",
    "\n",
    "# x - 2y >= 2\n",
    "y = (x - 2)/2\n",
    "plt.plot(x, y)\n",
    "plt.fill_between(x, y, y + 10, alpha=0.2)\n",
    "\n",
    "# y <= -2\n",
    "y = x * 0 - 2\n",
    "plt.plot(x, y)\n",
    "plt.fill_between(x, y, y - 10, alpha=0.2)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "După cum se vede și în grafic, nu există o zonă în care toate cele 3 semiplane să se intersecteze simultan. Deci nu avem unde să căutăm valoarea maximă a lui $x + y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sistem cu soluție optimă finită"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\max x + y \\\\\n",
    "x - 2y \\geq -5 \\\\\n",
    "x + y \\leq 3\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-5, 5, 200)\n",
    "\n",
    "plt.figure(dpi=100)\n",
    "\n",
    "plt.xlim(-3, 5)\n",
    "plt.ylim(-4, 4)\n",
    "\n",
    "# x - 2y >= -5\n",
    "y = (x + 5)/2\n",
    "plt.plot(x, y)\n",
    "plt.fill_between(x, y, y -10, alpha=0.2)\n",
    "\n",
    "# x + y <= 3\n",
    "y = 3 - x\n",
    "plt.plot(x, y)\n",
    "plt.fill_between(x, y, y - 10, alpha=0.2)\n",
    "\n",
    "# Punct de optim\n",
    "plt.scatter(1/3, 8/3, c='red', zorder=5)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cea mai mare valoare a lui $x + y$ se poate obține în punctul de intersecție ale dreptelor $x - 2y = 5$ și $x + y = 3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sistem cu optim unic, infinit\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\max x + y \\\\\n",
    "x + y <= 5\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-5, 5, 200)\n",
    "\n",
    "plt.figure(dpi=100)\n",
    "\n",
    "plt.xlim(-5, 5)\n",
    "plt.ylim(-5, 5)\n",
    "\n",
    "\n",
    "# x + y <= 5\n",
    "y = 5 - x\n",
    "plt.plot(x, y)\n",
    "plt.fill_between(x, y, y - 15, alpha=0.2)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observăm că toate punctele de pe dreapta $x + y = 5$ maximizează funcția obiectiv $x + y$, deci avem o infinitate de soluții optime."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sistem cu optim infinit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\max x + y \\\\\n",
    "x \\geq 0\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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eKQSmMTsjcEuSe5K8P8lZG59LkzHGz5Y32Uq5Jslnq+q2JN/Lxsuuzkty01KnWl03JnlXkrcmOVlVm/dmnBhjPLa8sVbXGONkkl/6gl9VjyT5+aqFQCIGWE2XJXn5bNkaXLX4cVbPGOMLVfX8JB9M8uJsfNJ8yxjjJ8udbGVtvoTzli3rr8jGDcrwG3GZAACa82oCAGhODABAc2IAAJoTAwDQnBgAgObEAAA0JwYAoDkxAADNiQEAaE4MAEBzYgAAmvs/Nryp3EQdsFQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=100)\n",
    "\n",
    "plt.xlim(-5, 5)\n",
    "plt.ylim(-5, 5)\n",
    "\n",
    "# x >= 0\n",
    "y = np.linspace(-5, 5, 200)\n",
    "plt.plot(0 * y, y)\n",
    "plt.fill_betweenx(y, 0, 5, alpha=0.2)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Putem să luăm orice pereche de puncte $(x, y)$ cu $x \\geq 0$, și putem obține o valoare oricât de mare vrem pentru funcția obiectiv $x + y$."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}