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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
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       "ry"
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
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      "text/latex": [
       "$\\displaystyle {===================================}$"
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{l}\n",
       "\\begin{array}{lcrcrcrcrcrcrcl}\n",
       " \\max \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} 10 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 14 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 12 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 18 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 20 x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 16 x_{6} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{6} \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{6} \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 8 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 5 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{6} \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 20 \\\\\n",
       "\\end{array} \\\\\n",
       "x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} \\geq 0\n",
       "\\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{l}\n",
       "\\begin{array}{lcrcrcrcrcrcrcl}\n",
       " \\max \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} 10 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 14 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 12 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 18 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 20 x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 16 x_{6} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{6} \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{6} \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 8 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 5 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{5} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{6} \\mspace{-6mu}&\\mspace{-6mu} = \\mspace{-6mu}&\\mspace{-6mu} 20 \\\\\n",
       "\\end{array} \\\\\n",
       "x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} \\geq 0\n",
       "\\end{array}$"
      ],
      "text/plain": [
       "LP problem (use 'view(...)' or '%display typeset' for details)"
      ]
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 104\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 104$"
      ],
      "text/plain": [
       "104"
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(0,\\,0,\\,2,\\,0,\\,4,\\,0\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(0,\\,0,\\,2,\\,0,\\,4,\\,0\\right)$"
      ],
      "text/plain": [
       "(0, 0, 2, 0, 4, 0)"
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
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      "text/latex": [
       "$\\displaystyle {===================================}$"
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       "ry"
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
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      "text/latex": [
       "$\\displaystyle {===================================}$"
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       "ry"
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    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {}$"
      ],
      "text/plain": [
       "salto"
      ]
     },
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    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {\\texttt{El dual es:}}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {\\texttt{El dual es:}}$"
      ],
      "text/plain": [
       "dual"
      ]
     },
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    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
      "text/plain": [
       "ry"
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{l}\n",
       "\\begin{array}{lcrcrcrcrcl}\n",
       " \\min \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} 6 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 8 y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 6 y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 20 y_{4} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 14 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 12 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 5 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 18 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 20 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 16 \\\\\n",
       "\\end{array} \\\\\n",
       "\n",
       "\\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{l}\n",
       "\\begin{array}{lcrcrcrcrcl}\n",
       " \\min \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} 6 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 8 y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 6 y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 20 y_{4} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 14 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 12 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 5 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 18 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 20 \\\\\n",
       " \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu}  \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{4} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 16 \\\\\n",
       "\\end{array} \\\\\n",
       "\n",
       "\\end{array}$"
      ],
      "text/plain": [
       "LP problem (use 'view(...)' or '%display typeset' for details)"
      ]
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
      "text/plain": [
       "ry"
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    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 104\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 104$"
      ],
      "text/plain": [
       "104"
      ]
     },
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    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(0,\\,-10,\\,-16,\\,14\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(0,\\,-10,\\,-16,\\,14\\right)$"
      ],
      "text/plain": [
       "(0, -10, -16, 14)"
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     },
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     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {}$"
      ],
      "text/plain": [
       "salto"
      ]
     },
     "metadata": {},
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    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
      "text/plain": [
       "ry"
      ]
     },
     "metadata": {},
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    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {\\ \\ \\ Fase\\ I}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {\\ \\ \\ Fase\\ I}$"
      ],
      "text/plain": [
       "FI"
      ]
     },
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     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
      "text/plain": [
       "ry"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {\\color{red}{t}}_{1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 6 \\\\\n",
       "{\\color{red}{t}}_{2} & 1 & 1 & 0 & 3 & 2 & 1 & 0 & 1 & 0 & 0 & 8 \\\\\n",
       "{\\color{red}{t}}_{3} & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 6 \\\\\n",
       "{\\color{red}{t}}_{4} & 3 & 4 & 2 & 5 & 4 & 2 & 0 & 0 & 0 & 1 & 20 \\\\\n",
       "\\hline\n",
       " {W_{ind}} & -6 & -8 & -4 & -10 & -8 & -4 & 0 & 0 & 0 & 0 & 40\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {\\color{red}{t}}_{1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 6 \\\\\n",
       "{\\color{red}{t}}_{2} & 1 & 1 & 0 & 3 & 2 & 1 & 0 & 1 & 0 & 0 & 8 \\\\\n",
       "{\\color{red}{t}}_{3} & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 6 \\\\\n",
       "{\\color{red}{t}}_{4} & 3 & 4 & 2 & 5 & 4 & 2 & 0 & 0 & 0 & 1 & 20 \\\\\n",
       "\\hline\n",
       " {W_{ind}} & -6 & -8 & -4 & -10 & -8 & -4 & 0 & 0 & 0 & 0 & 40\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[  B| x1  x2  x3  x4  x5  x6  t1  t2  t3  t4|rhs]\n",
       "[---+---------------------------------------+---]\n",
       "[ t1|  1   1   1   1   1   1   1   0   0   0|  6]\n",
       "[ t2|  1   1   0   3   2   1   0   1   0   0|  8]\n",
       "[ t3|  1   2   1   1   1   0   0   0   1   0|  6]\n",
       "[ t4|  3   4   2   5   4   2   0   0   0   1| 20]\n",
       "[---+---------------------------------------+---]\n",
       "[  W| -6  -8  -4 -10  -8  -4   0   0   0   0| 40]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cambio( 2 , 4 )\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {\\color{red}{t}}_{1} & \\frac{2}{3} & \\frac{2}{3} & 1 & 0 & \\frac{1}{3} & \\frac{2}{3} & 1 & -\\frac{1}{3} & 0 & 0 & \\frac{10}{3} \\\\\n",
       "{x}_{4} & \\frac{1}{3} & \\frac{1}{3} & 0 & 1 & \\frac{2}{3} & \\frac{1}{3} & 0 & \\frac{1}{3} & 0 & 0 & \\frac{8}{3} \\\\\n",
       "{\\color{red}{t}}_{3} & \\frac{2}{3} & \\frac{5}{3} & 1 & 0 & \\frac{1}{3} & -\\frac{1}{3} & 0 & -\\frac{1}{3} & 1 & 0 & \\frac{10}{3} \\\\\n",
       "{\\color{red}{t}}_{4} & \\frac{4}{3} & \\frac{7}{3} & 2 & 0 & \\frac{2}{3} & \\frac{1}{3} & 0 & -\\frac{5}{3} & 0 & 1 & \\frac{20}{3} \\\\\n",
       "\\hline\n",
       " {W_{ind}} & -\\frac{8}{3} & -\\frac{14}{3} & -4 & 0 & -\\frac{4}{3} & -\\frac{2}{3} & 0 & \\frac{10}{3} & 0 & 0 & \\frac{40}{3}\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {\\color{red}{t}}_{1} & \\frac{2}{3} & \\frac{2}{3} & 1 & 0 & \\frac{1}{3} & \\frac{2}{3} & 1 & -\\frac{1}{3} & 0 & 0 & \\frac{10}{3} \\\\\n",
       "{x}_{4} & \\frac{1}{3} & \\frac{1}{3} & 0 & 1 & \\frac{2}{3} & \\frac{1}{3} & 0 & \\frac{1}{3} & 0 & 0 & \\frac{8}{3} \\\\\n",
       "{\\color{red}{t}}_{3} & \\frac{2}{3} & \\frac{5}{3} & 1 & 0 & \\frac{1}{3} & -\\frac{1}{3} & 0 & -\\frac{1}{3} & 1 & 0 & \\frac{10}{3} \\\\\n",
       "{\\color{red}{t}}_{4} & \\frac{4}{3} & \\frac{7}{3} & 2 & 0 & \\frac{2}{3} & \\frac{1}{3} & 0 & -\\frac{5}{3} & 0 & 1 & \\frac{20}{3} \\\\\n",
       "\\hline\n",
       " {W_{ind}} & -\\frac{8}{3} & -\\frac{14}{3} & -4 & 0 & -\\frac{4}{3} & -\\frac{2}{3} & 0 & \\frac{10}{3} & 0 & 0 & \\frac{40}{3}\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[    B|   x1    x2    x3    x4    x5    x6    t1    t2    t3    t4|  rhs]\n",
       "[-----+-----------------------------------------------------------+-----]\n",
       "[   t1|  2/3   2/3     1     0   1/3   2/3     1  -1/3     0     0| 10/3]\n",
       "[   x4|  1/3   1/3     0     1   2/3   1/3     0   1/3     0     0|  8/3]\n",
       "[   t3|  2/3   5/3     1     0   1/3  -1/3     0  -1/3     1     0| 10/3]\n",
       "[   t4|  4/3   7/3     2     0   2/3   1/3     0  -5/3     0     1| 20/3]\n",
       "[-----+-----------------------------------------------------------+-----]\n",
       "[    W| -8/3 -14/3    -4     0  -4/3  -2/3     0  10/3     0     0| 40/3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cambio( 3 , 2 )\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {\\color{red}{t}}_{1} & \\frac{2}{5} & 0 & \\frac{3}{5} & 0 & \\frac{1}{5} & \\frac{4}{5} & 1 & -\\frac{1}{5} & -\\frac{2}{5} & 0 & 2 \\\\\n",
       "{x}_{4} & \\frac{1}{5} & 0 & -\\frac{1}{5} & 1 & \\frac{3}{5} & \\frac{2}{5} & 0 & \\frac{2}{5} & -\\frac{1}{5} & 0 & 2 \\\\\n",
       "{x}_{2} & \\frac{2}{5} & 1 & \\frac{3}{5} & 0 & \\frac{1}{5} & -\\frac{1}{5} & 0 & -\\frac{1}{5} & \\frac{3}{5} & 0 & 2 \\\\\n",
       "{\\color{red}{t}}_{4} & \\frac{2}{5} & 0 & \\frac{3}{5} & 0 & \\frac{1}{5} & \\frac{4}{5} & 0 & -\\frac{6}{5} & -\\frac{7}{5} & 1 & 2 \\\\\n",
       "\\hline\n",
       " {W_{ind}} & -\\frac{4}{5} & 0 & -\\frac{6}{5} & 0 & -\\frac{2}{5} & -\\frac{8}{5} & 0 & \\frac{12}{5} & \\frac{14}{5} & 0 & 4\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {\\color{red}{t}}_{1} & \\frac{2}{5} & 0 & \\frac{3}{5} & 0 & \\frac{1}{5} & \\frac{4}{5} & 1 & -\\frac{1}{5} & -\\frac{2}{5} & 0 & 2 \\\\\n",
       "{x}_{4} & \\frac{1}{5} & 0 & -\\frac{1}{5} & 1 & \\frac{3}{5} & \\frac{2}{5} & 0 & \\frac{2}{5} & -\\frac{1}{5} & 0 & 2 \\\\\n",
       "{x}_{2} & \\frac{2}{5} & 1 & \\frac{3}{5} & 0 & \\frac{1}{5} & -\\frac{1}{5} & 0 & -\\frac{1}{5} & \\frac{3}{5} & 0 & 2 \\\\\n",
       "{\\color{red}{t}}_{4} & \\frac{2}{5} & 0 & \\frac{3}{5} & 0 & \\frac{1}{5} & \\frac{4}{5} & 0 & -\\frac{6}{5} & -\\frac{7}{5} & 1 & 2 \\\\\n",
       "\\hline\n",
       " {W_{ind}} & -\\frac{4}{5} & 0 & -\\frac{6}{5} & 0 & -\\frac{2}{5} & -\\frac{8}{5} & 0 & \\frac{12}{5} & \\frac{14}{5} & 0 & 4\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[   B|  x1   x2   x3   x4   x5   x6   t1   t2   t3   t4| rhs]\n",
       "[----+-------------------------------------------------+----]\n",
       "[  t1| 2/5    0  3/5    0  1/5  4/5    1 -1/5 -2/5    0|   2]\n",
       "[  x4| 1/5    0 -1/5    1  3/5  2/5    0  2/5 -1/5    0|   2]\n",
       "[  x2| 2/5    1  3/5    0  1/5 -1/5    0 -1/5  3/5    0|   2]\n",
       "[  t4| 2/5    0  3/5    0  1/5  4/5    0 -6/5 -7/5    1|   2]\n",
       "[----+-------------------------------------------------+----]\n",
       "[   W|-4/5    0 -6/5    0 -2/5 -8/5    0 12/5 14/5    0|   4]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cambio( 1 , 6 )\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{6} & \\frac{1}{2} & 0 & \\frac{3}{4} & 0 & \\frac{1}{4} & 1 & \\frac{5}{4} & -\\frac{1}{4} & -\\frac{1}{2} & 0 & \\frac{5}{2} \\\\\n",
       "{x}_{4} & 0 & 0 & -\\frac{1}{2} & 1 & \\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 1 \\\\\n",
       "{x}_{2} & \\frac{1}{2} & 1 & \\frac{3}{4} & 0 & \\frac{1}{4} & 0 & \\frac{1}{4} & -\\frac{1}{4} & \\frac{1}{2} & 0 & \\frac{5}{2} \\\\\n",
       "{\\color{red}{t}}_{4} & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & 0 \\\\\n",
       "\\hline\n",
       " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{6} & \\frac{1}{2} & 0 & \\frac{3}{4} & 0 & \\frac{1}{4} & 1 & \\frac{5}{4} & -\\frac{1}{4} & -\\frac{1}{2} & 0 & \\frac{5}{2} \\\\\n",
       "{x}_{4} & 0 & 0 & -\\frac{1}{2} & 1 & \\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 1 \\\\\n",
       "{x}_{2} & \\frac{1}{2} & 1 & \\frac{3}{4} & 0 & \\frac{1}{4} & 0 & \\frac{1}{4} & -\\frac{1}{4} & \\frac{1}{2} & 0 & \\frac{5}{2} \\\\\n",
       "{\\color{red}{t}}_{4} & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & 0 \\\\\n",
       "\\hline\n",
       " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[   B|  x1   x2   x3   x4   x5   x6   t1   t2   t3   t4| rhs]\n",
       "[----+-------------------------------------------------+----]\n",
       "[  x6| 1/2    0  3/4    0  1/4    1  5/4 -1/4 -1/2    0| 5/2]\n",
       "[  x4|   0    0 -1/2    1  1/2    0 -1/2  1/2    0    0|   1]\n",
       "[  x2| 1/2    1  3/4    0  1/4    0  1/4 -1/4  1/2    0| 5/2]\n",
       "[  t4|   0    0    0    0    0    0   -1   -1   -1    1|   0]\n",
       "[----+-------------------------------------------------+----]\n",
       "[   W|   0    0    0    0    0    0    2    2    2    0|   0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "El problema de la fase I es óptimo y Wopt=0. Por tanto, el original es factible\n",
      "Preparemos el cuadro inicial de la fase II\n",
      "1\n",
      "[0, 0, 0, 1]\n",
      "6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
      "text/plain": [
       "ry"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {\\ \\ \\ Fase\\ II}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {\\ \\ \\ Fase\\ II}$"
      ],
      "text/plain": [
       "FII"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle {===================================}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle {===================================}$"
      ],
      "text/plain": [
       "ry"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{6} & \\frac{1}{2} & 0 & \\frac{3}{4} & 0 & \\frac{1}{4} & 1 & \\frac{5}{2} \\\\\n",
       "{x}_{4} & 0 & 0 & -\\frac{1}{2} & 1 & \\frac{1}{2} & 0 & 1 \\\\\n",
       "{x}_{2} & \\frac{1}{2} & 1 & \\frac{3}{4} & 0 & \\frac{1}{4} & 0 & \\frac{5}{2} \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -5 & 0 & -\\frac{3}{2} & 0 & \\frac{7}{2} & 0 & 93\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{6} & \\frac{1}{2} & 0 & \\frac{3}{4} & 0 & \\frac{1}{4} & 1 & \\frac{5}{2} \\\\\n",
       "{x}_{4} & 0 & 0 & -\\frac{1}{2} & 1 & \\frac{1}{2} & 0 & 1 \\\\\n",
       "{x}_{2} & \\frac{1}{2} & 1 & \\frac{3}{4} & 0 & \\frac{1}{4} & 0 & \\frac{5}{2} \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -5 & 0 & -\\frac{3}{2} & 0 & \\frac{7}{2} & 0 & 93\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[   B|  x1   x2   x3   x4   x5   x6| rhs]\n",
       "[----+-----------------------------+----]\n",
       "[  x6| 1/2    0  3/4    0  1/4    1| 5/2]\n",
       "[  x4|   0    0 -1/2    1  1/2    0|   1]\n",
       "[  x2| 1/2    1  3/4    0  1/4    0| 5/2]\n",
       "[----+-----------------------------+----]\n",
       "[   Z|  -5    0 -3/2    0  7/2    0|  93]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cambio( 2 , 5 )\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{6} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n",
       "{x}_{5} & 0 & 0 & -1 & 2 & 1 & 0 & 2 \\\\\n",
       "{x}_{2} & \\frac{1}{2} & 1 & 1 & -\\frac{1}{2} & 0 & 0 & 2 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -5 & 0 & 2 & -7 & 0 & 0 & 100\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{6} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n",
       "{x}_{5} & 0 & 0 & -1 & 2 & 1 & 0 & 2 \\\\\n",
       "{x}_{2} & \\frac{1}{2} & 1 & 1 & -\\frac{1}{2} & 0 & 0 & 2 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -5 & 0 & 2 & -7 & 0 & 0 & 100\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[   B|  x1   x2   x3   x4   x5   x6| rhs]\n",
       "[----+-----------------------------+----]\n",
       "[  x6| 1/2    0    1 -1/2    0    1|   2]\n",
       "[  x5|   0    0   -1    2    1    0|   2]\n",
       "[  x2| 1/2    1    1 -1/2    0    0|   2]\n",
       "[----+-----------------------------+----]\n",
       "[   Z|  -5    0    2   -7    0    0| 100]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cambio( 1 , 3 )\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n",
       "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & 4 \\\\\n",
       "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -6 & 0 & 0 & -6 & 0 & -2 & 104\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n",
       "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & 4 \\\\\n",
       "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -6 & 0 & 0 & -6 & 0 & -2 & 104\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[   B|  x1   x2   x3   x4   x5   x6| rhs]\n",
       "[----+-----------------------------+----]\n",
       "[  x3| 1/2    0    1 -1/2    0    1|   2]\n",
       "[  x5| 1/2    0    0  3/2    1    1|   4]\n",
       "[  x2|   0    1    0    0    0   -1|   0]\n",
       "[----+-----------------------------+----]\n",
       "[   Z|  -6    0    0   -6    0   -2| 104]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "El problema es óptimo\n"
     ]
    }
   ],
   "source": [
    "tipo='max'  # max o min\n",
    "\n",
    "n=6 #  número de variables x's\n",
    "m=4 #  número de restricciones (distintas de las de signo)\n",
    "\n",
    "c=matrix(QQ,1,n,[10,14,12,18,20,16] )  # vector de costos\n",
    "\n",
    "A=matrix(QQ,m,n,[   # coeficientes de las variables x's\n",
    "[1,1,1,1,1,1],\n",
    "[1,1,0,3,2,1],\n",
    "[1,2,1,1,1,0],\n",
    "[3,4,2,5,4,2]\n",
    "])\n",
    "\n",
    "b=matrix(QQ,m,1,[6,8,6,20])  # vector de la derecha\n",
    "\n",
    "\n",
    "######################################################\n",
    "############### VARIABLES DE HOLGURA  ################\n",
    "\n",
    "\n",
    "h=matrix(ZZ,1,m,[0,0,0,0])  # variables de holgura: \n",
    "                            # Pon 1 si va sumando, \n",
    "                            # -1 si va restando y \n",
    "                            # 0 si no hay variable de holgura.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "####################################################################\n",
    "############  HASTA AQUÍ LOS DATOS  ################################\n",
    "####################################################################\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "load('https://sage.unex.es/501708/simplex2023Auto.sage')\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7f09020",
   "metadata": {},
   "source": [
    "Veamos cómo cambia la solución del problema si cambiamos el coeficiente de costo de alguna de sus variables, por ejemplo, la $x_1$\n",
    "\n",
    "Para ello resolvemos el problema paramétrico $\n",
    "\\mathcal{P(\\lambda)}: \\quad max \\;\\; z_{\\lambda}(x)= c^t x + \\lambda c_{1p}^t x,\\quad \\text{s.a.:} \\quad Ax=b, \\quad x\\geq 0\n",
    "$\n",
    "\n",
    "para $c_{1p}=(1,0,0,0,0,0)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7c54b7d7",
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n",
       "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & 4 \\\\\n",
       "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -6 & 0 & 0 & -6 & 0 & -2 & 104 \\\\\n",
       "{Z{1p}} & 1 & 0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n",
       "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & 4 \\\\\n",
       "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & -6 & 0 & 0 & -6 & 0 & -2 & 104 \\\\\n",
       "{Z{1p}} & 1 & 0 & 0 & 0 & 0 & 0 & 0\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[   B|  x1   x2   x3   x4   x5   x6| rhs]\n",
       "[----+-----------------------------+----]\n",
       "[  x3| 1/2    0    1 -1/2    0    1|   2]\n",
       "[  x5| 1/2    0    0  3/2    1    1|   4]\n",
       "[  x2|   0    1    0    0    0   -1|   0]\n",
       "[----+-----------------------------+----]\n",
       "[   Z|  -6    0    0   -6    0   -2| 104]\n",
       "[ Z1p|   1    0    0    0    0    0|   0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "c1p=matrix(QQ,1,n,[1,0,0,0,0,0]); param1c(c1p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b6d065e",
   "metadata": {},
   "source": [
    "¿Para qué valores del parámetro $\\lambda$ es este cuadro óptimo?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81e1cb96",
   "metadata": {},
   "source": [
    "Resolvamos el sistema de inecuaciones: \n",
    "$$\n",
    "-6 + \\lambda \\leq 0 \\\\\n",
    "-6  \\leq 0 \\\\\n",
    "-2 \\leq 0 \n",
    "$$\n",
    "\n",
    "Su solución es $\\lambda\\in (-\\infty,6].$ Por tanto, el cuadro es óptimo para los valores de $\\lambda$ en dicho intervalo. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6688aeff",
   "metadata": {},
   "source": [
    "Si quisiéramos resolver problemas $\\mathcal{P}(\\lambda)$ para valores del parámetro a la derecha de 6, tendríamos que utilizar el método símplex: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c48a9e2e",
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{1} & 1 & 0 & 2 & -1 & 0 & 2 & 4 \\\\\n",
       "{x}_{5} & 0 & 0 & -1 & 2 & 1 & 0 & 2 \\\\\n",
       "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & 0 & 0 & 12 & -12 & 0 & 10 & 80 \\\\\n",
       "{Z{1p}} & 0 & 0 & -2 & 1 & 0 & -2 & 4\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{r|rrrrrr|r}\n",
       "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n",
       "\\hline\n",
       " {x}_{1} & 1 & 0 & 2 & -1 & 0 & 2 & 4 \\\\\n",
       "{x}_{5} & 0 & 0 & -1 & 2 & 1 & 0 & 2 \\\\\n",
       "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n",
       "\\hline\n",
       " {Z_{ind}} & 0 & 0 & 12 & -12 & 0 & 10 & 80 \\\\\n",
       "{Z{1p}} & 0 & 0 & -2 & 1 & 0 & -2 & 4\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[  B| x1  x2  x3  x4  x5  x6|rhs]\n",
       "[---+-----------------------+---]\n",
       "[ x1|  1   0   2  -1   0   2|  4]\n",
       "[ x5|  0   0  -1   2   1   0|  2]\n",
       "[ x2|  0   1   0   0   0  -1|  0]\n",
       "[---+-----------------------+---]\n",
       "[  Z|  0   0  12 -12   0  10| 80]\n",
       "[Z1p|  0   0  -2   1   0  -2|  4]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cambio(1,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7baad470",
   "metadata": {},
   "source": [
    "Resolviendo el siguiente sistema de inecuaciones, descubrimos que éste cuadro resuelve el problema paramétrico $\\mathcal{P}(\\lambda)$ si\n",
    "$$\n",
    "12 -2\\lambda \\leq 0 \\\\\n",
    "-12 + \\lambda \\leq 0 \\\\\n",
    "10 - 2\\lambda \\leq 0 \n",
    "$$\n",
    "esto es, si $$\\lambda\\in [6,12],$$ el cuadro anterior es óptimo para dicho problema."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c75199c6",
   "metadata": {},
   "source": [
    "etc"
   ]
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   "execution_count": null,
   "id": "20ba50c0",
   "metadata": {
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