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& 4 \\\\\n", "{\\color{red}{t}}_{3} & \\frac{1}{2} & \\frac{3}{2} & 1 & -\\frac{1}{2} & 0 & -\\frac{1}{2} & 0 & -\\frac{1}{2} & 1 & 0 & 2 \\\\\n", "{\\color{red}{t}}_{4} & 1 & -6 & 2 & -11 & 0 & 0 & 0 & -2 & 0 & 1 & 4 \\\\\n", "\\hline\n", " {W_{ind}} & -2 & 4 & -4 & 12 & 0 & 0 & 0 & 4 & 0 & 0 & 8\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[----+-------------------------------------------------+----]\n", "[ t1| 1/2 1/2 1 -1/2 0 1/2 1 -1/2 0 0| 2]\n", "[ x5| 1/2 1/2 0 3/2 1 1/2 0 1/2 0 0| 4]\n", "[ t3| 1/2 3/2 1 -1/2 0 -1/2 0 -1/2 1 0| 2]\n", "[ t4| 1 -6 2 -11 0 0 0 -2 0 1| 4]\n", "[----+-------------------------------------------------+----]\n", "[ W| -2 4 -4 12 0 0 0 4 0 0| 8]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 1 , 3 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & \\frac{1}{2} & 1 & -\\frac{1}{2} & 0 & \\frac{1}{2} & 1 & -\\frac{1}{2} & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & \\frac{1}{2} & 0 & \\frac{3}{2} & 1 & \\frac{1}{2} & 0 & \\frac{1}{2} & 0 & 0 & 4 \\\\\n", "{\\color{red}{t}}_{3} & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 0 \\\\\n", "{\\color{red}{t}}_{4} & 0 & -7 & 0 & -10 & 0 & -1 & -2 & -1 & 0 & 1 & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 6 & 0 & 10 & 0 & 2 & 4 & 2 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\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}_{3} & \\frac{1}{2} & \\frac{1}{2} & 1 & -\\frac{1}{2} & 0 & \\frac{1}{2} & 1 & -\\frac{1}{2} & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & \\frac{1}{2} & 0 & \\frac{3}{2} & 1 & \\frac{1}{2} & 0 & \\frac{1}{2} & 0 & 0 & 4 \\\\\n", "{\\color{red}{t}}_{3} & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 0 \\\\\n", "{\\color{red}{t}}_{4} & 0 & -7 & 0 & -10 & 0 & -1 & -2 & -1 & 0 & 1 & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 6 & 0 & 10 & 0 & 2 & 4 & 2 & 0 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[----+-------------------------------------------------+----]\n", "[ x3| 1/2 1/2 1 -1/2 0 1/2 1 -1/2 0 0| 2]\n", "[ x5| 1/2 1/2 0 3/2 1 1/2 0 1/2 0 0| 4]\n", "[ t3| 0 1 0 0 0 -1 -1 0 1 0| 0]\n", "[ t4| 0 -7 0 -10 0 -1 -2 -1 0 1| 0]\n", "[----+-------------------------------------------------+----]\n", "[ W| 0 6 0 10 0 2 4 2 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "El problema de la fase I es óptimo y Wopt=0. Por tanto, el original es factible\n", "Preparemos el cuadro inicial de la fase II\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & \\frac{3}{2} & -\\frac{1}{2} & -\\frac{1}{2} & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & \\frac{1}{2} & \\frac{1}{2} & -\\frac{1}{2} & 0 & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 0 \\\\\n", "{\\color{red}{t}}_{4} & 0 & 0 & 0 & -10 & 0 & -8 & -9 & -1 & 7 & 1 & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 10 & 0 & 8 & 10 & 2 & -6 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & 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"[----+-------------------------------------------------+----]\n", "[ W| 0 0 0 10 0 8 10 2 -6 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 1 & 1 & -\\frac{1}{2} & 0 & 0 & \\frac{1}{2} & -\\frac{1}{2} & \\frac{1}{2} & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 1 & 0 & \\frac{3}{2} & 1 & 0 & -\\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2} & 0 & 4 \\\\\n", "{x}_{6} & 0 & -1 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & 0 & 0 \\\\\n", "{\\color{red}{t}}_{4} & 0 & -8 & 0 & -10 & 0 & 0 & -1 & -1 & -1 & 1 & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 8 & 0 & 10 & 0 & 0 & 2 & 2 & 2 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ 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"data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ \\ \\ Fase\\ II}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ \\ \\ Fase\\ II}$$" ], "text/plain": [ "FII" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & \\frac{7}{5} & 1 & 0 & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & -\\frac{1}{5} & 0 & 0 & 1 & 0 & 4 \\\\\n", "{x}_{6} & 0 & -1 & 0 & 0 & 0 & 1 & 0 \\\\\n", "{x}_{4} & 0 & \\frac{4}{5} & 0 & 1 & 0 & 0 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & \\frac{14}{5} & 0 & 0 & 0 & 0 & 104\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & \\frac{7}{5} & 1 & 0 & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & -\\frac{1}{5} & 0 & 0 & 1 & 0 & 4 \\\\\n", "{x}_{6} & 0 & -1 & 0 & 0 & 0 & 1 & 0 \\\\\n", "{x}_{4} & 0 & \\frac{4}{5} & 0 & 1 & 0 & 0 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & \\frac{14}{5} & 0 & 0 & 0 & 0 & 104\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6| rhs]\n", "[----+-----------------------------+----]\n", "[ x3| 1/2 7/5 1 0 0 0| 2]\n", "[ x5| 1/2 -1/5 0 0 1 0| 4]\n", "[ x6| 0 -1 0 0 0 1| 0]\n", "[ x4| 0 4/5 0 1 0 0| 0]\n", "[----+-----------------------------+----]\n", "[ Z| -6 14/5 0 0 0 0| 104]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 4 , 2 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{7}{4} & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{7}{4} & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6| rhs]\n", "[----+-----------------------------+----]\n", "[ x3| 1/2 0 1 -7/4 0 0| 2]\n", "[ x5| 1/2 0 0 1/4 1 0| 4]\n", "[ x6| 0 0 0 5/4 0 1| 0]\n", "[ x2| 0 1 0 5/4 0 0| 0]\n", "[----+-----------------------------+----]\n", "[ Z| -6 0 0 -7/2 0 0| 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", "\n", "load('https://sage.unex.es/501708/simplex2023Auto.sage')\n" ] }, { "cell_type": "markdown", "id": "f7f09020", "metadata": {}, "source": [ "Calculemos el 'shadow price' para la fila 2. \n", "\n", "Para ello resolvemos el problema paramétrico \n", "\n", "$\n", "\\mathcal{P(\\lambda)}: \\quad max \\;\\; \\text{z(x)},\\quad \\text{s.a.:} \\quad Ax=b+\\lambda b_{1p}, \\quad x\\geq 0, \\quad\n", "$ \n", "\n", "para $\\lambda = 1$" ] }, { "cell_type": "code", "execution_count": 4, "id": "7c54b7d7", "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|rr}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} & {b_1} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{7}{4} & 0 & 0 & 2 & -\\frac{5}{8} \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 & \\frac{3}{8} \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 & \\frac{1}{8} \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 & \\frac{1}{8} \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104 & \\frac{15}{4}\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|rr}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} & {b_1} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{7}{4} & 0 & 0 & 2 & -\\frac{5}{8} \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 & \\frac{3}{8} \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 & \\frac{1}{8} \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 & \\frac{1}{8} \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104 & \\frac{15}{4}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6| rhs rhs1]\n", "[----+-----------------------------+---------]\n", "[ x3| 1/2 0 1 -7/4 0 0| 2 -5/8]\n", "[ x5| 1/2 0 0 1/4 1 0| 4 3/8]\n", "[ x6| 0 0 0 5/4 0 1| 0 1/8]\n", "[ x2| 0 1 0 5/4 0 0| 0 1/8]\n", "[----+-----------------------------+---------]\n", "[ Z| -6 0 0 -7/2 0 0| 104 15/4]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "b1p=matrix(QQ,m,1,[0,1,0,0]); param1b(b1p) " ] }, { "cell_type": "code", "execution_count": 5, "id": "2ab231ca", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[ NO NO NO 2.0000 NO NO]" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cocientesD(1)" ] }, { "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", "$$\n", "2 - 5/8\\lambda \\geq 0 \\\\\n", "4 + 3/8\\lambda \\geq 0 \\\\\n", "0 + 1/8\\lambda \\geq 0 \\\\\n", "0 + 1/8\\lambda \\geq 0 \n", "$$\n", "\n", "Su solución es $\\lambda\\in [0,16/5].$ Por tanto, el cuadro es óptimo para los valores de $\\lambda$ en dicho intervalo. " ] }, { "cell_type": "markdown", "id": "50d123b2", "metadata": {}, "source": [ "En particular, este cuadro es óptimo para el problema $\\mathcal{\\lambda = 1}.$ El valor de la función objetivo óptima es $104+15/4\\lambda$, con $\\lambda =1.$ Si la función objetivo $z$ representa un beneficio, la mejora al aumentar $b_2$ de 8 a 9 es 15/4. Ese es el \"precio en la sombra\" para esta línea 2. " ] }, { "cell_type": "markdown", "id": "6688aeff", "metadata": {}, "source": [ "Si quisiéramos resolver problemas $\\mathcal{P}(\\lambda)$ para valores del parámetro a la derecha de 16/5, tendríamos que utilizar el método símplex dual. " ] }, { "cell_type": "code", "execution_count": 2, "id": "c48a9e2e", "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{4} & -\\frac{2}{7} & 0 & -\\frac{4}{7} & 1 & 0 & 0 & -\\frac{8}{7} \\\\\n", "{x}_{5} & \\frac{4}{7} & 0 & \\frac{1}{7} & 0 & 1 & 0 & \\frac{30}{7} \\\\\n", "{x}_{6} & \\frac{5}{14} & 0 & \\frac{5}{7} & 0 & 0 & 1 & \\frac{10}{7} \\\\\n", "{x}_{2} & \\frac{5}{14} & 1 & \\frac{5}{7} & 0 & 0 & 0 & \\frac{10}{7} \\\\\n", "\\hline\n", " {Z_{ind}} & -7 & 0 & -2 & 0 & 0 & 0 & 108\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{4} & -\\frac{2}{7} & 0 & -\\frac{4}{7} & 1 & 0 & 0 & -\\frac{8}{7} \\\\\n", "{x}_{5} & \\frac{4}{7} & 0 & \\frac{1}{7} & 0 & 1 & 0 & \\frac{30}{7} \\\\\n", "{x}_{6} & \\frac{5}{14} & 0 & \\frac{5}{7} & 0 & 0 & 1 & \\frac{10}{7} \\\\\n", "{x}_{2} & \\frac{5}{14} & 1 & \\frac{5}{7} & 0 & 0 & 0 & \\frac{10}{7} \\\\\n", "\\hline\n", " {Z_{ind}} & -7 & 0 & -2 & 0 & 0 & 0 & 108\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6| rhs]\n", "[----+-----------------------------+----]\n", "[ x4|-2/7 0 -4/7 1 0 0|-8/7]\n", "[ x5| 4/7 0 1/7 0 1 0|30/7]\n", "[ x6|5/14 0 5/7 0 0 1|10/7]\n", "[ x2|5/14 1 5/7 0 0 0|10/7]\n", "[----+-----------------------------+----]\n", "[ Z| -7 0 -2 0 0 0| 108]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(1,4)" ] }, { "cell_type": "markdown", "id": "7baad470", "metadata": {}, "source": [ "Resolviendo un siguiente sistema de inecuaciones obtenido de manera similar al alterior, encontraremos los valores del parámetro \\lambda para los que ese cuadro es óptimo." ] }, { "cell_type": "markdown", "id": "c75199c6", "metadata": {}, "source": [ "etc" ] }, { "cell_type": "code", "execution_count": null, "id": "ed732fbe", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.5", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.2" } }, "nbformat": 4, "nbformat_minor": 5 }