vaje2-LinearniProgrami.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "# Linearni programi in Sage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "## Razporeditev sredstev\n",
    "\n",
    "Podjetje ima tri oddelke, ki poslujejo z: 30% dobičkom, 20% dobičkom in brez dobička. Med oddelke želimo razporediti 40.000 evrov sredstev, pri čemer mora 1. oddelek prejeti vsaj 8.000 evrov, 2. oddelek vsaj 10.000 evrov in 3. oddelek največ dvakrat toliko kot 2. oddelek. Kako naj razporedimo sredstva, da bomo dobili čimveč dobička?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.0"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = MixedIntegerLinearProgram(maximization=True)\n",
    "x = p.new_variable(real=True)\n",
    "p.set_objective(0.3 * x[1] + 0.2 * x[2])\n",
    "p.add_constraint(x[1] >= 8)\n",
    "p.add_constraint(x[2] >= 10)\n",
    "p.add_constraint(x[3] >= 0)\n",
    "p.add_constraint(x[3] <= 2 * x[2])\n",
    "p.add_constraint(x[1] + x[2] + x[3] <= 40)\n",
    "p.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 30.0, 2: 10.0, 3: 0.0}"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.get_values(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Grafično reševanje\n",
    "\n",
    "Grafično in z `MixedIntegerLinearProgram` rešite linearni problem, podan z\n",
    "\n",
    "\\begin{align*}\n",
    "\\min x + y \\\\\n",
    "x + 2y &\\ge 1 \\\\\n",
    "2x + y &\\ge 1 \\\\\n",
    "2x - y &\\le 1\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "nastavitve = {'plot_points': 1000, 'incol': 'lightblue', 'bordercol': 'gray'}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0dd735c28068482da2b1382bf866f0c5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function _ at 0x7fde996a65e0> with 1 widget\n",
       "  k: TransformFloatSlider(value=0.5, descrip…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = var('x y')\n",
    "meje = [(x, 0, 1), (y, 0, 1)]\n",
    "rp = region_plot([x + 2*y >= 1, 2*x + y >= 1, 2*x - y <= 1], *meje, **nastavitve)\n",
    "\n",
    "@interact\n",
    "def _(k=slider(0, 3, default=0.5, step_size=0.01, label='$k$')):\n",
    "    show(rp + implicit_plot(x + y - k, *meje))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6666666666666667"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = MixedIntegerLinearProgram(maximization=False)\n",
    "v = p.new_variable(real=True)\n",
    "p.set_objective(v[x] + v[y])\n",
    "p.add_constraint(v[x] + 2 * v[y] >= 1)\n",
    "p.add_constraint(2 * v[x] + v[y] >= 1)\n",
    "p.add_constraint(2 * v[x] - v[y] <= 1)\n",
    "p.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: 0.3333333333333333, y: 0.33333333333333337}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.get_values(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Proizvodnja fotoaparatov\n",
    "\n",
    "Fotografski zanesenjak Niko N. je tik pred prazniki ustanovil s.p., v katerem bo proizvajal dve vrsti fotoaparatov: Minimum 3 in Optimum 7. Za vsak aparat Minimum 3 potrebuje 3 ure dela in material, vreden 70 evrov, za vsak Optimum 7 pa za 180 evrov materiala in 5 ur dela. Optimum 7 lahko proda za 300 evrov, Minimum 3 pa za 130 evrov. Katere fotoaparate naj naredi Niko, če ima 2100 evrov začetnega kapitala, s katerim lahko nakupi material, do praznikov pa ima le še 71 ur časa?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1560.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = MixedIntegerLinearProgram(maximization=True)\n",
    "v = p.new_variable(real=True)\n",
    "p.set_objective((300 - 180) * v['opt'] + (130 - 70) * v['min'])\n",
    "p.add_constraint(5 * v['opt'] + 3 * v['min'] <= 71)\n",
    "p.add_constraint(180 * v['opt'] + 70 * v['min'] <= 2100)\n",
    "p.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'opt': 7.000000000000002, 'min': 11.999999999999998}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.get_values(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Analiza proizvodnje fotoaparatov\n",
    "\n",
    "Nikov problem poskusite rešiti še grafično. Ko ga rešite, spreminjajte parametre in opazujte, kako se s tem spreminja množica dopustnih rešitev ter naklon grafa kriterijske funkcije."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "7bbe0d45f9784e5690809a00448aadb2",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Manual interactive function <function _ at 0x7fde964ef310> with 9 widgets\n",
       "  optimumUre: TransformFloatSlider(v…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = var('x y')\n",
    "meje = [(x, 0, 20), (y, 0, 30)]\n",
    "\n",
    "@interact\n",
    "def _(optimumUre=slider(0, 10, default=5, step_size=0.01),\n",
    "      minimumUre=slider(0, 10, default=3, step_size=0.01),\n",
    "      optimumStrosek=slider(0, 300, default=180, step_size=0.5),\n",
    "      minimumStrosek=slider(0, 300, default=70, step_size=0.5),\n",
    "      optimumCena=slider(0, 1000, default=300, step_size=2),\n",
    "      minimumCena=slider(0, 1000, default=130, step_size=2),\n",
    "      kapital=slider(0, 5000, default=2100, step_size=10),\n",
    "      cas=slider(0, 200, default=71, step_size=0.5),\n",
    "      dobicek=slider(0, 3000, default=1560, step_size=5),\n",
    "      auto_update=False):\n",
    "    show(region_plot([optimumUre * x + minimumUre * y <= cas, optimumStrosek * x + minimumStrosek * y <= kapital],\n",
    "                     *meje, **nastavitve) +\n",
    "         implicit_plot((optimumCena - optimumStrosek) * x + (minimumCena - minimumStrosek) * y - dobicek, *meje) +\n",
    "         text(\"\"\"\n",
    "max %.2f opt + %.2f min  p.p.\n",
    "        %.2f opt + %.2f min <= %.2f\n",
    "        %.2f opt + %.2f min <= %.2f\n",
    "              \"\"\" % (optimumCena - optimumStrosek, minimumCena - minimumStrosek,\n",
    "                     optimumUre, minimumUre, cas, optimumStrosek, minimumStrosek, kapital),\n",
    "              (3, 29), axes=False, horizontal_alignment='left'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sestavljanje jedilnika\n",
    "\n",
    "Z jedilnikom, sestavljenim iz živil v spodnji tabeli želimo zadostiti dnevnim potrebam po vsaj 2000 kCal energije, 55 g beljakovin in 800 mg kalcija. Kaj naj pojemo, da bomo za živila porabili čim manj denarja, hkrati pa ne bomo jedli preveč enolične hrane (glej zadnji stolpec)?\n",
    "\n",
    "| živilo    | porcija    | energija (kCal) | beljakovine (g) | kalcij (mg) | cena (€) | max. št. porcij |\n",
    "| --------- | --------   | --------------- | --------------- | ----------- | -------- | --------------- |\n",
    "| kaša      | 28 g       | 110             | 4               | 2           | 3        | 4               |\n",
    "| piščanec\t| 100 g      | 205             | 32              | 12          | 24       | 3               |\n",
    "| jajca     | 2 kom.     | 160             | 13              | 54          | 13       | 2               |\n",
    "| mleko     | 237 cm$^3$ | 160             | 8               | 285         | 9        | 8               |\n",
    "| kremšnita | 170 g\t     | 420             | 4               | 22          | 20       | 2               |\n",
    "| pasulj    | 260 g      | 260             | 14              | 80          | 19       | 2               |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "92.5"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = MixedIntegerLinearProgram(maximization=False)\n",
    "v = p.new_variable(real=True)\n",
    "p.set_objective(3 * v['kaša'] + 24 * v['piščanec'] + 13 * v['jajca'] +\n",
    "                9 * v['mleko'] + 20 * v['kremšnita'] + 19 * v['pasulj'])\n",
    "p.add_constraint(110 * v['kaša'] + 205 * v['piščanec'] + 160 * v['jajca'] +\n",
    "                 160 * v['mleko'] + 420 * v['kremšnita'] + 260 * v['pasulj'] >= 2000)\n",
    "p.add_constraint(4 * v['kaša'] + 32 * v['piščanec'] + 13 * v['jajca'] +\n",
    "                 8 * v['mleko'] + 4 * v['kremšnita'] + 14 * v['pasulj'] >= 55)\n",
    "p.add_constraint(2 * v['kaša'] + 12 * v['piščanec'] + 54 * v['jajca'] +\n",
    "                 285 * v['mleko'] + 22 * v['kremšnita'] + 80 * v['pasulj'] >= 800)\n",
    "p.add_constraint(0 <= v['kaša'] <= 4)\n",
    "p.add_constraint(0 <= v['piščanec'] <= 3)\n",
    "p.add_constraint(0 <= v['jajca'] <= 2)\n",
    "p.add_constraint(0 <= v['mleko'] <= 8)\n",
    "p.add_constraint(0 <= v['kremšnita'] <= 2)\n",
    "p.add_constraint(0 <= v['pasulj'] <= 2)\n",
    "p.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'kaša': 4.0,\n",
       " 'piščanec': 0.0,\n",
       " 'jajca': 0.0,\n",
       " 'mleko': 4.499999999999999,\n",
       " 'kremšnita': 2.0000000000000004,\n",
       " 'pasulj': 0.0}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.get_values(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Največja včrtana krožnica\n",
    "\n",
    "Poiščite največjo krožnico, vsebovano v večkotniku, podanem z enačbami:\n",
    "\n",
    "\\begin{align*}\n",
    "-x + y &\\le 4 \\\\\n",
    "y &\\le 5 \\\\\n",
    "2x + y &\\le 15 \\\\\n",
    "{x \\over 2} + y &\\ge 1\n",
    "\\end{align*}\n",
    "\n",
    "V pomoč naj vam bo, da je točka $x$ od hiperravnine z enačbo $a^\\top x = b$ oddaljena ${{b - a^\\top x} \\over {\\lVert a \\rVert}}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.6781472845828778"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = MixedIntegerLinearProgram(maximization=True)\n",
    "v = p.new_variable(real=True)\n",
    "r, x, y = (v[i] for i in \"rxy\")\n",
    "p.set_objective(r)\n",
    "p.add_constraint(r * sqrt(2) <= 4 + x - y)\n",
    "p.add_constraint(r * 1 <= 5 - y)\n",
    "p.add_constraint(r * sqrt(5) <= 15 - 2*x - y)\n",
    "p.add_constraint(r * sqrt(5)/2 <= -1 + x/2 + y)\n",
    "p.add_constraint(r >= 0)\n",
    "p.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'r': 2.6781472845828778, 'x': 3.3448139512495434, 'y': 2.3218527154171227}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = p.get_values(v)\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 7 graphics primitives"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y = var('x y')\n",
    "meje = [(x, -5, 10), (y, -5, 10)]\n",
    "sum((implicit_plot(eq, color=col, *meje) for eq, col in zip([-x + y - 4, y - 5, 2*x + y - 15, x/2 + y - 1], rainbow(4))),\n",
    "   region_plot([-x + y <= 4, y <= 5, 2*x + y <= 15, x/2 + y >= 1], *meje, **nastavitve) +\n",
    "   circle((res['x'], res['y']), res['r'], color='black'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Naftna rafinerija\n",
    "\n",
    "Rafinerija iz nafte tipov 1 in 2 meša bencin in kurilno olje. Na voljo ima 5000 sodčkov nafte tipa 1 s kvaliteto 10 in 10000 sodčkov nafte tipa 2 s kvaliteto 5. Povprečna kvaliteta bencina mora biti vsaj 8, povprečna kvaliteta kurilnega olja pa vsaj 6. Za vsak dolar reklame za kurilno olje proda 10 sodčkov kurilnega olja po ceni 20 dolarjev za sodček, za vsak dolar reklame za bencin pa 5 sodčkov bencina po ceni 25 dolarjev za sodček. Kako naj nameša nafto, da bo iztržila čimveč dobička?\n",
    "\n",
    "* $x_{1,b}$ ... število sodčkov tipa 1 za bencin\n",
    "* $x_{2,b}$ ... število sodčkov tipa 2 za bencin\n",
    "* $x_{1,k}$ ... število sodčkov tipa 1 za kurilno olje\n",
    "* $x_{2,k}$ ... število sodčkov tipa 2 za kurilno olje\n",
    "\n",
    "* Povprečna kvaliteta bencina: ${10 x_{1,b} + 5 x_{2,b} \\over x_{1,b} + x_{2,b}} \\le 8$\n",
    "* Povprečna kvaliteta kurlinega olja: ${10 x_{1,k} + 5 x_{2,k} \\over x_{1,k} + x_{2,k}} \\le 6$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "323000.0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = MixedIntegerLinearProgram(maximization=True)\n",
    "bencin = p.new_variable(real=True)\n",
    "kurilno_olje = p.new_variable(real=True)\n",
    "p.set_objective((25 - 1/5) * (bencin[1] + bencin[2]) + (20 - 1/10) * (kurilno_olje[1] + kurilno_olje[2]))\n",
    "p.add_constraint(bencin[1] + kurilno_olje[1] <= 5000)\n",
    "p.add_constraint(bencin[2] + kurilno_olje[2] <= 10000)\n",
    "p.add_constraint(10 * bencin[1] + 5 * bencin[2] >= 8 * (bencin[1] + bencin[2]))\n",
    "p.add_constraint(10 * kurilno_olje[1] + 5 * kurilno_olje[2] >= 6 * (kurilno_olje[1] + kurilno_olje[2]))\n",
    "p.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 3000.0000000000005, 2: 2000.0000000000002}"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.get_values(bencin)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 1999.9999999999993, 2: 7999.999999999999}"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.get_values(kurilno_olje)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Mešanje tonikov\n",
    "\n",
    "Na voljo imamo zdravilne tonike A, B in C z vsebnostjo zdravilnh učinkovin I, II in III, podano v spodnji tabeli. Z mešanjem tonikov želimo dobiti napitek, ki bo imel vsebnosti učinkovin, kot so navedene v zadnji vrstici tabele. Natančnih vrednosti ni moč doseči, zato za mero kvalitete mešanice vzamemo po absolutni vrednosti največje odstopanje posamezne učinkovine od predpisane vrednosti (v mešanici, ki vsebuje enak delež vsakega tonika, je ta vrednost enaka 2/3). Kakšni so deleži v najboljši mešanici?\n",
    "\n",
    "| tonik  |  I | II | III |\n",
    "| ------ | -- | -- | --- |\n",
    "| A      | 10 |  2 |   5 |\n",
    "| B      | 15 |  4 |   2 |\n",
    "| C      | 20 |  3 |   3 |\n",
    "| iskani | 15 |  3 |   4 |\n",
    "\n",
    "* $x_1, x_2, x_3 \\ge 0$\n",
    "* $x_1 + x_2 + x_3 = 1$\n",
    "* $\\min \\max\\{|10x_1 + 15x_2 + 20x_3 - 15|, |2x_1 + 4x_2 + 3x_3 - 3|, |5x_1 + 2x_2 + 3x_3 - 4|\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.2777777777777777,\n",
       " {'a': 0.4444444444444446,\n",
       "  'b': 0.1666666666666669,\n",
       "  'c': 0.3888888888888886,\n",
       "  'z': 1.0})"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spremenljivke = 'abcz'\n",
    "koeficienti = [[10, 15, 20, -15],\n",
    "                [2,  4,  3,  -3],\n",
    "                [5,  2,  3,  -4]]\n",
    "\n",
    "def izraz(koef, znak, v):\n",
    "    return sum(znak * x * v[k] for x, k in zip(koef, spremenljivke))\n",
    "               \n",
    "def resitev(koef, znak):\n",
    "    p = MixedIntegerLinearProgram(maximization=False)\n",
    "    v = p.new_variable(real=True)\n",
    "    ciljna = izraz(koef, znak, v)\n",
    "    p.set_objective(ciljna)\n",
    "    p.add_constraint(sum(v[k] for k in spremenljivke[:3]) == 1)\n",
    "    p.add_constraint(v[spremenljivke[-1]] == 1)\n",
    "    for k in spremenljivke:\n",
    "        p.add_constraint(v[k] >= 0)\n",
    "    for z in [-1, 1]:\n",
    "        for r in koeficienti:\n",
    "            p.add_constraint(ciljna >= izraz(r, z, v))\n",
    "    try:\n",
    "        opt = p.solve()\n",
    "    except sage.numerical.mip.MIPSolverException:\n",
    "        return (Infinity, None)\n",
    "    return (opt, p.get_values(v))\n",
    "\n",
    "min((resitev(koef, znak) for znak in [-1, 1] for koef in koeficienti), key=lambda t: t[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "ef84afa082554990bb636999d8eb0278",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function _ at 0x7fde963ba1f0> with 1 widget\n",
       "  k: TransformFloatSlider(value=1.0, descrip…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a, b, c, z = spr = var(' '.join(spremenljivke))\n",
    "razlike = [sum(k*x for k, x in zip(koef, spr)).subs(z == 1, c == 1 - a - b) for koef in koeficienti]\n",
    "meje = [(a, 0, 1), (b, 0, 1)]\n",
    "nast = {'plot_points': 200, 'bordercol': 'gray'}\n",
    "\n",
    "pogoji = [[a + b <= 1] + [znak * ciljna >= z * r for z in [-1, 1] for r in razlike if (z, r) != (znak, ciljna)]\n",
    "          for znak in [-1, 1] for ciljna in razlike]\n",
    "obmocja = sum(region_plot(pogoj, incol=barva, *meje, **nast) for pogoj, barva in zip(pogoji, rainbow(6)))\n",
    "\n",
    "@interact\n",
    "def _(k=slider(0, 1, default=1, step_size=0.01)):\n",
    "    warnings.filterwarnings(\"ignore\")\n",
    "    show(obmocja + region_plot([znak * ciljna <= k for znak in [-1, 1] for ciljna in razlike],\n",
    "                               incol='black', alpha=0.5, bordercol='magenta', borderwidth=3, plot_points=200, *meje))"
   ]
  }
 ],
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