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{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{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} & \\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)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[----+-------------------------------------------------+----]\n", "[ x3| 1/2 1/2 1 -1/2 0 1/2 1 -1/2 0 0| 2]\n", "[ x5| 1/2 1/2 0 3/2 1 1/2 0 1/2 0 0| 4]\n", "[ t3| 0 1 0 0 0 -1 -1 0 1 0| 0]\n", "[ t4| 0 -7 0 -10 0 -1 -2 -1 0 1| 0]\n", "[----+-------------------------------------------------+----]\n", "[ W| 0 6 0 10 0 2 4 2 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "El problema de la fase I es óptimo y Wopt=0. Por tanto, el original es factible\n", "Preparemos el cuadro inicial de la fase II\n", "2\n", "[0, 0, 1, 1]\n", "6\n", "cambio( 3 , 2 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & \\frac{3}{2} & -\\frac{1}{2} & -\\frac{1}{2} & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & \\frac{1}{2} & \\frac{1}{2} & -\\frac{1}{2} & 0 & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 0 \\\\\n", "{\\color{red}{t}}_{4} & 0 & 0 & 0 & -10 & 0 & -8 & -9 & -1 & 7 & 1 & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 10 & 0 & 8 & 10 & 2 & -6 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {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)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[----+-------------------------------------------------+----]\n", "[ x3| 1/2 0 1 -1/2 0 1 3/2 -1/2 -1/2 0| 2]\n", "[ x5| 1/2 0 0 3/2 1 1 1/2 1/2 -1/2 0| 4]\n", "[ x2| 0 1 0 0 0 -1 -1 0 1 0| 0]\n", "[ t4| 0 0 0 -10 0 -8 -9 -1 7 1| 0]\n", "[----+-------------------------------------------------+----]\n", "[ W| 0 0 0 10 0 8 10 2 -6 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "[0, 0, 0, 1]\n", "6\n", "cambio( 4 , 4 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & 0 & 0 & \\frac{7}{5} & \\frac{39}{20} & -\\frac{9}{20} & -\\frac{17}{20} & -\\frac{1}{20} & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & 0 & 1 & -\\frac{1}{5} & -\\frac{17}{20} & \\frac{7}{20} & \\frac{11}{20} & \\frac{3}{20} & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 0 \\\\\n", "{x}_{4} & 0 & 0 & 0 & 1 & 0 & \\frac{4}{5} & \\frac{9}{10} & \\frac{1}{10} & -\\frac{7}{10} & -\\frac{1}{10} & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\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} & 0 & 1 & 0 & 0 & \\frac{7}{5} & \\frac{39}{20} & -\\frac{9}{20} & -\\frac{17}{20} & -\\frac{1}{20} & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & 0 & 1 & -\\frac{1}{5} & -\\frac{17}{20} & \\frac{7}{20} & \\frac{11}{20} & \\frac{3}{20} & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & 0 \\\\\n", "{x}_{4} & 0 & 0 & 0 & 1 & 0 & \\frac{4}{5} & \\frac{9}{10} & \\frac{1}{10} & -\\frac{7}{10} & -\\frac{1}{10} & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[------+---------------------------------------------------------------------+------]\n", "[ x3| 1/2 0 1 0 0 7/5 39/20 -9/20 -17/20 -1/20| 2]\n", "[ x5| 1/2 0 0 0 1 -1/5 -17/20 7/20 11/20 3/20| 4]\n", "[ x2| 0 1 0 0 0 -1 -1 0 1 0| 0]\n", "[ x4| 0 0 0 1 0 4/5 9/10 1/10 -7/10 -1/10| 0]\n", "[------+---------------------------------------------------------------------+------]\n", "[ W| 0 0 0 0 0 0 1 1 1 1| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "[0, 0, 0, 0]\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ \\ \\ Fase\\ II}$ " ], "text/latex": [ "$$\\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}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & 0 & 0 & \\frac{7}{5} & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & 0 & 1 & -\\frac{1}{5} & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n", "{x}_{4} & 0 & 0 & 0 & 1 & 0 & \\frac{4}{5} & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & 0 & 0 & \\frac{14}{5} & 104\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\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 & 0 & 0 & \\frac{7}{5} & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & 0 & 1 & -\\frac{1}{5} & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n", "{x}_{4} & 0 & 0 & 0 & 1 & 0 & \\frac{4}{5} & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & 0 & 0 & \\frac{14}{5} & 104\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6| rhs]\n", "[----+-----------------------------+----]\n", "[ x3| 1/2 0 1 0 0 7/5| 2]\n", "[ x5| 1/2 0 0 0 1 -1/5| 4]\n", "[ x2| 0 1 0 0 0 -1| 0]\n", "[ x4| 0 0 0 1 0 4/5| 0]\n", "[----+-----------------------------+----]\n", "[ Z| -6 0 0 0 0 14/5| 104]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 4 , 6 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{7}{4} & 0 & 0 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\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}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 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", "[ x2| 0 1 0 5/4 0 0| 0]\n", "[ x6| 0 0 0 5/4 0 1| 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": "code", "execution_count": 7, "id": "c8e81e51", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{3}{8} & -\\frac{5}{8} & \\frac{3}{8} & \\frac{1}{8} \\\\\n", "-\\frac{5}{8} & \\frac{3}{8} & \\frac{3}{8} & \\frac{1}{8} \\\\\n", "\\frac{9}{8} & \\frac{1}{8} & -\\frac{7}{8} & -\\frac{1}{8} \\\\\n", "\\frac{1}{8} & \\frac{1}{8} & \\frac{1}{8} & -\\frac{1}{8}\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{3}{8} & -\\frac{5}{8} & \\frac{3}{8} & \\frac{1}{8} \\\\\n", "-\\frac{5}{8} & \\frac{3}{8} & \\frac{3}{8} & \\frac{1}{8} \\\\\n", "\\frac{9}{8} & \\frac{1}{8} & -\\frac{7}{8} & -\\frac{1}{8} \\\\\n", "\\frac{1}{8} & \\frac{1}{8} & \\frac{1}{8} & -\\frac{1}{8}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 3/8 -5/8 3/8 1/8]\n", "[-5/8 3/8 3/8 1/8]\n", "[ 9/8 1/8 -7/8 -1/8]\n", "[ 1/8 1/8 1/8 -1/8]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(BM1)" ] }, { "cell_type": "markdown", "id": "31286f11", "metadata": {}, "source": [ "# Problema b paramétrico" ] }, { "cell_type": "markdown", "id": "f7f09020", "metadata": {}, "source": [ "Resolvamos el siguiente problema paramétrico, en el que $b_p$ es $\\left( \\begin{array}{c}1\\\\-1\\\\0\\\\2 \\end{array}\n", "\\right)$. \n", " \n", "\n", "$\n", "\\mathcal{P(\\lambda)}: \\quad max \\;\\; \\text{z(x)},\\quad \\text{s.a.:} \\quad Ax=b+\\lambda b_{p}, \\quad x\\geq 0, \\quad\n", "$ \n", "\n", "para $\\lambda \\in \\mathbb R$" ] }, { "cell_type": "code", "execution_count": 8, "id": "29763803", "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|rr}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} & {b_1} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{7}{4} & 0 & 0 & 2 & \\frac{5}{4} \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 & -\\frac{3}{4} \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 & \\frac{3}{4} \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 & -\\frac{1}{4} \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104 & \\frac{17}{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\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}{4} \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{1}{4} & 1 & 0 & 4 & -\\frac{3}{4} \\\\\n", "{x}_{6} & 0 & 0 & 0 & \\frac{5}{4} & 0 & 1 & 0 & \\frac{3}{4} \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{5}{4} & 0 & 0 & 0 & -\\frac{1}{4} \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -\\frac{7}{2} & 0 & 0 & 104 & \\frac{17}{2}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6| rhs rhs1]\n", "[----+-----------------------------+---------]\n", "[ x3| 1/2 0 1 -7/4 0 0| 2 5/4]\n", "[ x5| 1/2 0 0 1/4 1 0| 4 -3/4]\n", "[ x6| 0 0 0 5/4 0 1| 0 3/4]\n", "[ x2| 0 1 0 5/4 0 0| 0 -1/4]\n", "[----+-----------------------------+---------]\n", "[ Z| -6 0 0 -7/2 0 0| 104 17/2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "bp=matrix(QQ,m,1,[1,-1,0,2]); param1b(bp) " ] }, { "cell_type": "markdown", "id": "7b6d065e", "metadata": {}, "source": [ "¿Para qué valores del parámetro $\\lambda$ es este cuadro óptimo?" ] }, { "cell_type": "markdown", "id": "81e1cb96", "metadata": {}, "source": [ "Resolvamos el sistema de inecuaciones: \n", "$$\n", "2 + 5/4\\lambda \\geq 0 \\\\\n", "4 - 3/4\\lambda \\geq 0 \\\\\n", "0 + 3/4\\lambda \\geq 0 \\\\\n", "0 - 1/4\\lambda \\geq 0 \n", "$$\n", "\n", "Su solución es $\\lambda\\in 0.$ Por tanto, la tabla es óptima para el problema $\\matcal P(\\lambda),$ si $\\lambda=0$. " ] }, { "cell_type": "markdown", "id": "f7ebd276", "metadata": {}, "source": [ "### El siguiente paso será resolver justo a la derecha de 0, esto es, en $0^+$" ] }, { "cell_type": "markdown", "id": "a3eae5a0", "metadata": {}, "source": [ "Para esos valores del parámetro $\\lambda$, todas las coordenadas de $b(\\lambda)$ son positivas, salvo la cuarta. Daremos entonces un paso del método símplex dual." ] }, { "cell_type": "code", "execution_count": null, "id": "6443f431", "metadata": {}, "outputs": [], "source": [] }, { "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 }