{ "cells": [ { "cell_type": "code", "execution_count": 5, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcrcrcl}\n", " \\max \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 10 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 20 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 40 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 32 x_{4} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 10 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 30 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 80 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 40 x_{4} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 900 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 x_{4} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 80 \\\\\n", "\\end{array} \\\\\n", "x_{1}, x_{2}, x_{3}, x_{4} \\geq 0\n", "\\end{array}$$" ], "text/plain": [ "LP problem (use 'view(...)' or '%display typeset' for details)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}740$$" ], "text/plain": [ "740" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(10,\\,0,\\,0,\\,20\\right)$$" ], "text/plain": [ "(10, 0, 0, 20)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$" ], "text/plain": [ "salto" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\texttt{El dual es:}}$$" ], "text/plain": [ "dual" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcl}\n", " \\min \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 900 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 80 y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 10 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 30 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 20 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 80 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 40 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 40 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 32 \\\\\n", "\\end{array} \\\\\n", "y_{1}, y_{2} \\geq 0\n", "\\end{array}$$" ], "text/plain": [ "LP problem (use 'view(...)' or '%display typeset' for details)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}740$$" ], "text/plain": [ "740" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{17}{25},\\,\\frac{8}{5}\\right)$$" ], "text/plain": [ "(17/25, 8/5)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$" ], "text/plain": [ "salto" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 10 & 30 & 80 & 40 & 1 & 0 & 900 \\\\\n", "{\\color{green}{s}}_{2} & 2 & 1 & 1 & 3 & 0 & 1 & 80 \\\\\n", "\\hline\n", " {Z_{ind}} & 10 & 20 & 40 & 32 & 0 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2|rhs]\n", "[---+-----------------------+---]\n", "[ s1| 10 30 80 40 1 0|900]\n", "[ s2| 2 1 1 3 0 1| 80]\n", "[---+-----------------------+---]\n", "[ Z| 10 20 40 32 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 1 , 3 )\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{3} & \\frac{1}{8} & \\frac{3}{8} & 1 & \\frac{1}{2} & \\frac{1}{80} & 0 & \\frac{45}{4} \\\\\n", "{\\color{green}{s}}_{2} & \\frac{15}{8} & \\frac{5}{8} & 0 & \\frac{5}{2} & -\\frac{1}{80} & 1 & \\frac{275}{4} \\\\\n", "\\hline\n", " {Z_{ind}} & 5 & 5 & 0 & 12 & -\\frac{1}{2} & 0 & 450\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2| rhs]\n", "[-----+-----------------------------------+-----]\n", "[ x3| 1/8 3/8 1 1/2 1/80 0| 45/4]\n", "[ s2| 15/8 5/8 0 5/2 -1/80 1|275/4]\n", "[-----+-----------------------------------+-----]\n", "[ Z| 5 5 0 12 -1/2 0| 450]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 1 , 4 )\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{4} & \\frac{1}{4} & \\frac{3}{4} & 2 & 1 & \\frac{1}{40} & 0 & \\frac{45}{2} \\\\\n", "{\\color{green}{s}}_{2} & \\frac{5}{4} & -\\frac{5}{4} & -5 & 0 & -\\frac{3}{40} & 1 & \\frac{25}{2} \\\\\n", "\\hline\n", " {Z_{ind}} & 2 & -4 & -24 & 0 & -\\frac{4}{5} & 0 & 720\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2| rhs]\n", "[-----+-----------------------------------+-----]\n", "[ x4| 1/4 3/4 2 1 1/40 0| 45/2]\n", "[ s2| 5/4 -5/4 -5 0 -3/40 1| 25/2]\n", "[-----+-----------------------------------+-----]\n", "[ Z| 2 -4 -24 0 -4/5 0| 720]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "cambio( 2 , 1 )\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{4} & 0 & 1 & 3 & 1 & \\frac{1}{25} & -\\frac{1}{5} & 20 \\\\\n", "{x}_{1} & 1 & -1 & -4 & 0 & -\\frac{3}{50} & \\frac{4}{5} & 10 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -2 & -16 & 0 & -\\frac{17}{25} & -\\frac{8}{5} & 740\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 s1 s2| rhs]\n", "[------+-----------------------------------------+------]\n", "[ x4| 0 1 3 1 1/25 -1/5| 20]\n", "[ x1| 1 -1 -4 0 -3/50 4/5| 10]\n", "[------+-----------------------------------------+------]\n", "[ Z| 0 -2 -16 0 -17/25 -8/5| 740]" ] }, "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=4 # número de variables x's\n", "m=2 # número de restricciones (distintas de las de signo)\n", "\n", "coste=matrix(QQ,1,n,[20,30,45,58] )\n", "venta=matrix(QQ,1,n,[30,50,85,90] )\n", "\n", "c=venta-coste # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[10,30,80,40],\n", "[2,1,1,3]\n", "])\n", "\n", "b=matrix(QQ,m,1,[900,80]) # vector de la derecha\n", "\n", "\n", "h=matrix(ZZ,1,m,[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", "\n", "\n", "load('https://sage.unex.es/501708/simplex2023Auto.sage')\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "\\frac{1}{25} & -\\frac{1}{5} \\\\\n", "-\\frac{3}{50} & \\frac{4}{5}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 1/25 -1/5]\n", "[-3/50 4/5]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(BM1)" ] } ], "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", 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