{ "cells": [ { "cell_type": "code", "execution_count": 20, "metadata": { "scrolled": true }, "outputs": [ { "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcl}\n", " \\max \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 15 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 12 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 11 \\\\\n", "\\end{array} \\\\\n", "x_{1}, x_{2} \\geq 0\n", "\\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcl}\n", " \\max \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 15 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 12 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 11 \\\\\n", "\\end{array} \\\\\n", "x_{1}, x_{2} \\geq 0\n", "\\end{array}$$" ], "text/plain": [ "LP problem (use 'view(...)' or '%display typeset' for details)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{27}{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{27}{2}$$" ], "text/plain": [ "27/2" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{7}{2},\\,\\frac{3}{2}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{7}{2},\\,\\frac{3}{2}\\right)$$" ], "text/plain": [ "(7/2, 3/2)" ] }, "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$" ], "text/plain": [ "salto" ] }, "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 1 & 4 & 1 & 0 & 0 & 0 & 15 \\\\\n", "{\\color{green}{s}}_{2} & 3 & 1 & 0 & 1 & 0 & 0 & 12 \\\\\n", "{\\color{green}{s}}_{3} & 2 & 2 & 0 & 0 & 1 & 0 & 10 \\\\\n", "{\\color{green}{s}}_{4} & 3 & -1 & 0 & 0 & 0 & 1 & 11 \\\\\n", "\\hline\n", " {Z_{ind}} & 3 & 2 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 1 & 4 & 1 & 0 & 0 & 0 & 15 \\\\\n", "{\\color{green}{s}}_{2} & 3 & 1 & 0 & 1 & 0 & 0 & 12 \\\\\n", "{\\color{green}{s}}_{3} & 2 & 2 & 0 & 0 & 1 & 0 & 10 \\\\\n", "{\\color{green}{s}}_{4} & 3 & -1 & 0 & 0 & 0 & 1 & 11 \\\\\n", "\\hline\n", " {Z_{ind}} & 3 & 2 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 s1 s2 s3 s4|rhs]\n", "[---+-----------------------+---]\n", "[ s1| 1 4 1 0 0 0| 15]\n", "[ s2| 3 1 0 1 0 0| 12]\n", "[ s3| 2 2 0 0 1 0| 10]\n", "[ s4| 3 -1 0 0 0 1| 11]\n", "[---+-----------------------+---]\n", "[ Z| 3 2 0 0 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tipo ='max' # Problema de tipo max o min\n", "\n", "n=2 # número de variables x's\n", "m=4 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[3,2] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[1,4],\n", "[3,1],\n", "[2,2],\n", "[3,-1]\n", "])\n", "\n", "b=matrix(QQ,m,1,[15,12,10,11]) # vector de la derecha\n", "\n", "\n", "h=matrix(ZZ,1,m,[1,1,1,1]) # 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,[0,0,0,0]) # 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=(3,4,5,6) # posición de la matriz identidad\n", "\n", "\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ LOS DATOS ################################\n", "####################################################################\n", "\n", "\n", "\n", "\n", "load('https://sage.unex.es/501708/simplex2023CompletamenteManual.sage')" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "cocientes(6)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "cambio(4,1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "cambio(2,2)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "cambio(3,6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ... A continuación otro problema ..." ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcl}\n", " \\min \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 8 \\\\\n", "\\end{array} \\\\\n", "x_{1} \\geq 0\n", "\\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcl}\n", " \\min \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 8 \\\\\n", "\\end{array} \\\\\n", "x_{1} \\geq 0\n", "\\end{array}$$" ], "text/plain": [ "LP problem (use 'view(...)' or '%display typeset' for details)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\infty$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\infty$$" ], "text/plain": [ "-Infinity" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{None}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{None}$$" ], "text/plain": [ "None" ] }, "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$" ], "text/plain": [ "salto" ] }, "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 3 & 2 & 1 & 0 & 6 \\\\\n", "{\\color{green}{s}}_{2} & 2 & 4 & 0 & 1 & 8 \\\\\n", "\\hline\n", " {Z_{ind}} & 2 & 1 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 3 & 2 & 1 & 0 & 6 \\\\\n", "{\\color{green}{s}}_{2} & 2 & 4 & 0 & 1 & 8 \\\\\n", "\\hline\n", " {Z_{ind}} & 2 & 1 & 0 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 s1 s2|rhs]\n", "[---+---------------+---]\n", "[ s1| 3 2 1 0| 6]\n", "[ s2| 2 4 0 1| 8]\n", "[---+---------------+---]\n", "[ Z| 2 1 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tipo='min' # Problema de tipo max o min\n", "\n", "n=2 # número de variables x's\n", "m=2 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[2,1] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[3,2],\n", "[2,4]\n", "])\n", "\n", "b=matrix(QQ,m,1,[6,8]) # 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", "t=matrix(ZZ,1,m,[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=(3,4) # posición de la matriz identidad\n", "\n", "signosvar=[\">=\", \"\"]\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ LOS DATOS ################################\n", "####################################################################\n", "\n", "\n", "\n", "\n", "load('https://sage.unex.es/501708/simplex2023CompletamenteManual2.sage')" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 2 & 0 & 1 & -\\frac{1}{2} & 2 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & 0 & \\frac{1}{4} & 2 \\\\\n", "\\hline\n", " {Z_{ind}} & \\frac{3}{2} & 0 & 0 & -\\frac{1}{4} & 2\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 2 & 0 & 1 & -\\frac{1}{2} & 2 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & 0 & \\frac{1}{4} & 2 \\\\\n", "\\hline\n", " {Z_{ind}} & \\frac{3}{2} & 0 & 0 & -\\frac{1}{4} & 2\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 s1 s2| rhs]\n", "[----+-------------------+----]\n", "[ s1| 2 0 1 -1/2| 2]\n", "[ x2| 1/2 1 0 1/4| 2]\n", "[----+-------------------+----]\n", "[ Z| 3/2 0 0 -1/4| 2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(2,2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ... A continuación otra versión de este problema ..." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "tipo='min' # Problema de tipo max o min\n", "\n", "n=3 # número de variables x's\n", "m=2 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[2,1, -1] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[3,2,-2],\n", "[2,4,-4]\n", "])\n", "\n", "b=matrix(QQ,m,1,[6,8]) # 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", "t=matrix(ZZ,1,m,[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=(4,5) # posición de la matriz identidad\n", "\n", "\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ LOS DATOS ################################\n", "####################################################################\n", "\n", "\n", "\n", "\n", "load('https://sage.unex.es/501708/simplex2023CompletamenteManual.sage')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Otro" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$" ], "text/plain": [ "ry" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcrcl}\n", " \\min \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 7 x_{3} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} 4 x_{3} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 8 \\\\\n", "\\end{array} \\\\\n", "x_{1}, x_{2}, x_{3} \\geq 0\n", "\\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n", "\\begin{array}{lcrcrcrcl}\n", " \\min \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 7 x_{3} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{2} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 6 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 4 x_{2} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} 4 x_{3} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 8 \\\\\n", "\\end{array} \\\\\n", "x_{1}, x_{2}, x_{3} \\geq 0\n", "\\end{array}$$" ], "text/plain": [ "LP problem (use 'view(...)' or '%display typeset' for details)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}3$$" ], "text/plain": [ "3" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,3,\\,0\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,3,\\,0\\right)$$" ], "text/plain": [ "(0, 3, 0)" ] }, "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$" ], "text/plain": [ "salto" ] }, "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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{red}{t}}_{1} & 3 & 2 & -2 & -1 & 0 & 1 & 0 & 6 \\\\\n", "{\\color{red}{t}}_{2} & 2 & 4 & -4 & 0 & -1 & 0 & 1 & 8 \\\\\n", "\\hline\n", " {W_{ind}} & -5 & -6 & 6 & 1 & 1 & 0 & 0 & 14\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{red}{t}}_{1} & 3 & 2 & -2 & -1 & 0 & 1 & 0 & 6 \\\\\n", "{\\color{red}{t}}_{2} & 2 & 4 & -4 & 0 & -1 & 0 & 1 & 8 \\\\\n", "\\hline\n", " {W_{ind}} & -5 & -6 & 6 & 1 & 1 & 0 & 0 & 14\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 s1 s2 t1 t2|rhs]\n", "[---+---------------------------+---]\n", "[ t1| 3 2 -2 -1 0 1 0| 6]\n", "[ t2| 2 4 -4 0 -1 0 1| 8]\n", "[---+---------------------------+---]\n", "[ W| -5 -6 6 1 1 0 0| 14]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tipo='min' # Problema de tipo max o min\n", "\n", "n=3 # número de variables x's\n", "m=2 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[2,1, -1+8] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[3,2,-2],\n", "[2,4,-4]\n", "])\n", "\n", "b=matrix(QQ,m,1,[6,8]) # 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", "t=matrix(ZZ,1,m,[1,1]) # 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=(6,7) # posición de la matriz identidad\n", "\n", "\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ LOS DATOS ################################\n", "####################################################################\n", "\n", "\n", "\n", "\n", "load('https://sage.unex.es/501708/simplex2023CompletamenteManual.sage')" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{red}{t}}_{1} & 2 & 0 & 0 & -1 & \\frac{1}{2} & 1 & -\\frac{1}{2} & 2 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & -1 & 0 & -\\frac{1}{4} & 0 & \\frac{1}{4} & 2 \\\\\n", "\\hline\n", " {W_{ind}} & -2 & 0 & 0 & 1 & -\\frac{1}{2} & 0 & \\frac{3}{2} & 2\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{red}{t}}_{1} & 2 & 0 & 0 & -1 & \\frac{1}{2} & 1 & -\\frac{1}{2} & 2 \\\\\n", "{x}_{2} & \\frac{1}{2} & 1 & -1 & 0 & -\\frac{1}{4} & 0 & \\frac{1}{4} & 2 \\\\\n", "\\hline\n", " {W_{ind}} & -2 & 0 & 0 & 1 & -\\frac{1}{2} & 0 & \\frac{3}{2} & 2\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 s1 s2 t1 t2| rhs]\n", "[----+----------------------------------+----]\n", "[ t1| 2 0 0 -1 1/2 1 -1/2| 2]\n", "[ x2| 1/2 1 -1 0 -1/4 0 1/4| 2]\n", "[----+----------------------------------+----]\n", "[ W| -2 0 0 1 -1/2 0 3/2| 2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(2,2)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{4} & \\frac{1}{2} & -\\frac{1}{4} & 1 \\\\\n", "{x}_{2} & 0 & 1 & -1 & \\frac{1}{4} & -\\frac{3}{8} & -\\frac{1}{4} & \\frac{3}{8} & \\frac{3}{2} \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{red}{t}}_{1} & {\\color{red}{t}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{4} & \\frac{1}{2} & -\\frac{1}{4} & 1 \\\\\n", "{x}_{2} & 0 & 1 & -1 & \\frac{1}{4} & -\\frac{3}{8} & -\\frac{1}{4} & \\frac{3}{8} & \\frac{3}{2} \\\\\n", "\\hline\n", " {W_{ind}} & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 s1 s2 t1 t2| rhs]\n", "[----+----------------------------------+----]\n", "[ x1| 1 0 0 -1/2 1/4 1/2 -1/4| 1]\n", "[ x2| 0 1 -1 1/4 -3/8 -1/4 3/8| 3/2]\n", "[----+----------------------------------+----]\n", "[ W| 0 0 0 0 0 1 1| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(1,1)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{4} & 1 \\\\\n", "{x}_{2} & 0 & 1 & -1 & \\frac{1}{4} & -\\frac{3}{8} & \\frac{3}{2} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 8 & \\frac{3}{4} & -\\frac{1}{8} & \\frac{7}{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {x}_{1} & 1 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{4} & 1 \\\\\n", "{x}_{2} & 0 & 1 & -1 & \\frac{1}{4} & -\\frac{3}{8} & \\frac{3}{2} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 8 & \\frac{3}{4} & -\\frac{1}{8} & \\frac{7}{2}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 s1 s2| rhs]\n", "[----+------------------------+----]\n", "[ x1| 1 0 0 -1/2 1/4| 1]\n", "[ x2| 0 1 -1 1/4 -3/8| 3/2]\n", "[----+------------------------+----]\n", "[ Z| 0 0 8 3/4 -1/8| 7/2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tt=(0,0); faseII(0,tt)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{2} & 4 & 0 & 0 & -2 & 1 & 4 \\\\\n", "{x}_{2} & \\frac{3}{2} & 1 & -1 & -\\frac{1}{2} & 0 & 3 \\\\\n", "\\hline\n", " {Z_{ind}} & \\frac{1}{2} & 0 & 8 & \\frac{1}{2} & 0 & 3\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{2} & 4 & 0 & 0 & -2 & 1 & 4 \\\\\n", "{x}_{2} & \\frac{3}{2} & 1 & -1 & -\\frac{1}{2} & 0 & 3 \\\\\n", "\\hline\n", " {Z_{ind}} & \\frac{1}{2} & 0 & 8 & \\frac{1}{2} & 0 & 3\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 s1 s2| rhs]\n", "[----+------------------------+----]\n", "[ s2| 4 0 0 -2 1| 4]\n", "[ x2| 3/2 1 -1 -1/2 0| 3]\n", "[----+------------------------+----]\n", "[ Z| 1/2 0 8 1/2 0| 3]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(1,5)" ] }, { "cell_type": "code", "execution_count": null, "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": 4 }