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derecha\n", "\n", "\n", "h=matrix(ZZ,1,m,[1,1,1]) # 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", "t=matrix(ZZ,1,m,[0,0,0]) # variables artificiales: \n", " # pon 1 si utilizas variable artificial en la fila i-ésima\n", " # 0 si no utilizas variable artificial en la fila i-ésima\n", "\n", "colId=(5,6,7) # posición de la matriz identidad\n", "\n", "\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ LOS DATOS ################################\n", "####################################################################\n", "\n", "\n", "#load('https://sage.unex.es/501708/simplex2023Auto.sage')\n", "\n", "load('https://sage.unex.es/501708/simplex2023CompletamenteManual.sage')" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ " 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-1/4 1/4 0| 0]\n", "[ s3| 0 0 -1/2 4 3/4 -11/4 1| 1]\n", "[-----+-----------------------------------------+-----]\n", "[ Z| 0 0 29/2 -98 -27/4 -53/4 0| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(2,2)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {x}_{3} & 2 & 0 & 1 & -8 & -\\frac{3}{2} & \\frac{11}{2} & 0 & 0 \\\\\n", "{x}_{2} & -1 & 1 & 0 & 2 & \\frac{1}{2} & -\\frac{5}{2} & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & -29 & 0 & 0 & 18 & 15 & -93 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & 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"cell_type": "markdown", "metadata": {}, "source": [ "Hemos vuelto a la tabla inicial! Apliquemos ahora las reglas de R.Bland. Suponemos que el orden establecido es el de los índices, primero las x's y luego las s's." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & -11 & -5 & 18 & 2 & 0 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{2} & 0 & 4 & 2 & -8 & -1 & 1 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 0 & 11 & 5 & -18 & -2 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 53 & 41 & -204 & -20 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & -11 & -5 & 18 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& -4 & -\\frac{3}{4} & \\frac{11}{4} & 0 & 0 \\\\\n", "{x}_{2} & 0 & 1 & \\frac{1}{2} & -2 & -\\frac{1}{4} & \\frac{1}{4} & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 0 & 0 & -\\frac{1}{2} & 4 & \\frac{3}{4} & -\\frac{11}{4} & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & \\frac{29}{2} & -98 & -\\frac{27}{4} & -\\frac{53}{4} & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & 0 & \\frac{1}{2} & -4 & -\\frac{3}{4} & \\frac{11}{4} & 0 & 0 \\\\\n", "{x}_{2} & 0 & 1 & \\frac{1}{2} & -2 & -\\frac{1}{4} & \\frac{1}{4} & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 0 & 0 & -\\frac{1}{2} & 4 & \\frac{3}{4} & -\\frac{11}{4} & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & \\frac{29}{2} & -98 & -\\frac{27}{4} & -\\frac{53}{4} & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2 s3| rhs]\n", "[-----+-----------------------------------------+-----]\n", "[ x1| 1 0 1/2 -4 -3/4 11/4 0| 0]\n", "[ x2| 0 1 1/2 -2 -1/4 1/4 0| 0]\n", "[ s3| 0 0 -1/2 4 3/4 -11/4 1| 1]\n", "[-----+-----------------------------------------+-----]\n", "[ Z| 0 0 29/2 -98 -27/4 -53/4 0| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(2,2)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {x}_{3} & 2 & 0 & 1 & -8 & -\\frac{3}{2} & \\frac{11}{2} & 0 & 0 \\\\\n", "{x}_{2} & -1 & 1 & 0 & 2 & \\frac{1}{2} & -\\frac{5}{2} & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", 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(distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[10,-57,-9,-24] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[1/2,-11/2,-5/2,9],\n", "[1/2,-3/2,-1/2,1],\n", "[1,0,0,0]\n", "])\n", "\n", "b=matrix(QQ,m,1,[0,0,1]) # vector de la derecha\n", "\n", "\n", "h=matrix(ZZ,1,m,[1,1,1]) # 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", "t=matrix(ZZ,1,m,[0,0,0]) # variables artificiales: \n", " # pon 1 si utilizas variable artificial en la fila i-ésima\n", " # 0 si no utilizas variable artificial en la fila i-ésima\n", "\n", "colId=(5,6,7) # posición de la matriz identidad\n", "\n", "\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ LOS DATOS ################################\n", "####################################################################\n", "\n", "\n", "#load('https://sage.unex.es/501708/simplex2023Auto.sage')\n", "\n", "load('https://sage.unex.es/501708/simplex2023CompletamenteManual.sage')" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & -11 & -5 & 18 & 2 & 0 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{2} & 0 & 4 & 2 & -8 & -1 & 1 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 0 & 11 & 5 & -18 & -2 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 53 & 41 & -204 & -20 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} 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{Z_{ind}} & 22 & -93 & -21 & 0 & 0 & 24 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & -4 & 8 & 2 & 0 & 1 & -9 & 0 & 0 \\\\\n", "{x}_{4} & \\frac{1}{2} & -\\frac{3}{2} & -\\frac{1}{2} & 1 & 0 & 1 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 22 & -93 & -21 & 0 & 0 & 24 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2 s3| rhs]\n", "[----+----------------------------------+----]\n", "[ s1| -4 8 2 0 1 -9 0| 0]\n", "[ x4| 1/2 -3/2 -1/2 1 0 1 0| 0]\n", "[ s3| 1 0 0 0 0 0 1| 1]\n", "[----+----------------------------------+----]\n", "[ Z| 22 -93 -21 0 0 24 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & \\frac{1}{2} & -\\frac{11}{2} & -\\frac{5}{2} & 9 & 1 & 0 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{2} & \\frac{1}{2} & -\\frac{3}{2} & -\\frac{1}{2} & 1 & 0 & 1 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 10 & -57 & -9 & -24 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & \\frac{1}{2} & -\\frac{11}{2} & -\\frac{5}{2} & 9 & 1 & 0 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{2} & 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$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & -2 & 2 & 0 & 4 & 1 & -5 & 0 & 0 \\\\\n", "{x}_{3} & -1 & 3 & 1 & -2 & 0 & -2 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 1 & -30 & 0 & -42 & 0 & -18 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & -2 & 2 & 0 & 4 & 1 & -5 & 0 & 0 \\\\\n", "{x}_{3} & -1 & 3 & 1 & -2 & 0 & -2 & 0 & 0 \\\\\n", "{\\color{green}{s}}_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 1 & -30 & 0 & -42 & 0 & -18 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2 s3|rhs]\n", "[---+---------------------------+---]\n", "[ s1| -2 2 0 4 1 -5 0| 0]\n", "[ x3| -1 3 1 -2 0 -2 0| 0]\n", "[ s3| 1 0 0 0 0 0 1| 1]\n", "[---+---------------------------+---]\n", "[ Z| 1 -30 0 -42 0 -18 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & 2 & 0 & 4 & 1 & -5 & 2 & 2 \\\\\n", "{x}_{3} & 0 & 3 & 1 & -2 & 0 & -2 & 1 & 1 \\\\\n", "{x}_{1} & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -30 & 0 & -42 & 0 & -18 & -1 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & 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