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{\\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)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[-----+-----------------------------------------------------------+-----]\n", "[ t1| 2/3 2/3 1 0 1/3 2/3 1 -1/3 0 0| 10/3]\n", "[ x4| 1/3 1/3 0 1 2/3 1/3 0 1/3 0 0| 8/3]\n", "[ t3| 2/3 5/3 1 0 1/3 -1/3 0 -1/3 1 0| 10/3]\n", "[ t4| 4/3 7/3 2 0 2/3 1/3 0 -5/3 0 1| 20/3]\n", "[-----+-----------------------------------------------------------+-----]\n", "[ W| -8/3 -14/3 -4 0 -4/3 -2/3 0 10/3 0 0| 40/3]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 3 , 2 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{red}{t}}_{1} & \\frac{2}{5} & 0 & \\frac{3}{5} & 0 & \\frac{1}{5} & \\frac{4}{5} & 1 & -\\frac{1}{5} & -\\frac{2}{5} & 0 & 2 \\\\\n", "{x}_{4} & \\frac{1}{5} & 0 & -\\frac{1}{5} & 1 & \\frac{3}{5} & \\frac{2}{5} & 0 & \\frac{2}{5} & -\\frac{1}{5} & 0 & 2 \\\\\n", "{x}_{2} & \\frac{2}{5} & 1 & \\frac{3}{5} & 0 & \\frac{1}{5} & -\\frac{1}{5} & 0 & -\\frac{1}{5} & \\frac{3}{5} & 0 & 2 \\\\\n", "{\\color{red}{t}}_{4} & \\frac{2}{5} & 0 & \\frac{3}{5} & 0 & \\frac{1}{5} & \\frac{4}{5} & 0 & -\\frac{6}{5} & -\\frac{7}{5} & 1 & 2 \\\\\n", "\\hline\n", " {W_{ind}} & -\\frac{4}{5} & 0 & -\\frac{6}{5} & 0 & -\\frac{2}{5} & -\\frac{8}{5} & 0 & \\frac{12}{5} & \\frac{14}{5} & 0 & 4\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\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", " {\\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)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[----+-------------------------------------------------+----]\n", "[ t1| 2/5 0 3/5 0 1/5 4/5 1 -1/5 -2/5 0| 2]\n", "[ x4| 1/5 0 -1/5 1 3/5 2/5 0 2/5 -1/5 0| 2]\n", "[ x2| 2/5 1 3/5 0 1/5 -1/5 0 -1/5 3/5 0| 2]\n", "[ t4| 2/5 0 3/5 0 1/5 4/5 0 -6/5 -7/5 1| 2]\n", "[----+-------------------------------------------------+----]\n", "[ W|-4/5 0 -6/5 0 -2/5 -8/5 0 12/5 14/5 0| 4]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 1 , 6 )\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {\\color{red}{t}}_{3} & {\\color{red}{t}}_{4} & {b} \\\\\n", "\\hline\n", " {x}_{6} & \\frac{1}{2} & 0 & \\frac{3}{4} & 0 & \\frac{1}{4} & 1 & \\frac{5}{4} & -\\frac{1}{4} & -\\frac{1}{2} & 0 & \\frac{5}{2} \\\\\n", "{x}_{4} & 0 & 0 & -\\frac{1}{2} & 1 & \\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 1 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & \\frac{3}{4} & 0 & \\frac{1}{4} & 0 & \\frac{1}{4} & -\\frac{1}{4} & \\frac{1}{2} & 0 & \\frac{5}{2} \\\\\n", "{\\color{red}{t}}_{4} & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & 0 \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\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}_{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)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 x6 t1 t2 t3 t4| rhs]\n", "[----+-------------------------------------------------+----]\n", "[ x6| 1/2 0 3/4 0 1/4 1 5/4 -1/4 -1/2 0| 5/2]\n", "[ x4| 0 0 -1/2 1 1/2 0 -1/2 1/2 0 0| 1]\n", "[ x2| 1/2 1 3/4 0 1/4 0 1/4 -1/4 1/2 0| 5/2]\n", "[ t4| 0 0 0 0 0 0 -1 -1 -1 1| 0]\n", "[----+-------------------------------------------------+----]\n", "[ W| 0 0 0 0 0 0 2 2 2 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "El problema de la fase I es óptimo y Wopt=0. Por tanto, el original es factible\n", "Preparemos el cuadro inicial de la fase II\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$ " ], "text/latex": [ "$$\\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}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " 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-5 & 0 & -\\frac{3}{2} & 0 & \\frac{7}{2} & 0 & 93\n", "\\end{array}\\right)$$" ], "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{6} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n", "{x}_{5} & 0 & 0 & -1 & 2 & 1 & 0 & 2 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & 1 & -\\frac{1}{2} & 0 & 0 & 2 \\\\\n", "\\hline\n", " {Z_{ind}} & -5 & 0 & 2 & -7 & 0 & 0 & 100\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{6} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n", "{x}_{5} & 0 & 0 & -1 & 2 & 1 & 0 & 2 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & 1 & -\\frac{1}{2} & 0 & 0 & 2 \\\\\n", "\\hline\n", " {Z_{ind}} & -5 & 0 & 2 & -7 & 0 & 0 & 100\n", "\\end{array}\\right)$$" ], "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -6 & 0 & -2 & 104\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {x}_{6} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{2} & 0 & 1 & -\\frac{1}{2} & 0 & 1 & 2 \\\\\n", "{x}_{5} & \\frac{1}{2} & 0 & 0 & \\frac{3}{2} & 1 & 1 & 4 \\\\\n", "{x}_{2} & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\\\\n", "\\hline\n", " {Z_{ind}} & -6 & 0 & 0 & -6 & 0 & -2 & 104\n", "\\end{array}\\right)$$" ], "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": 6, "id": "7c54b7d7", "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrr|}\n", "10 & 14 & 12 & 18 & 20 & 16\n", 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