{ "cells": [ { "cell_type": "code", "execution_count": 41, "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}{lcrcrcrcrcrcl}\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} + 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"{\\color{green}{s}}_{4} & 3 & -1 & -1 & -1 & 1 & 0 & 0 & 0 & 1 & 11 \\\\\n", "\\hline\n", " {Z_{ind}} & 3 & -2 & 1 & 2 & 3 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4|rhs]\n", "[---+-----------------------------------+---]\n", "[ s1| 1 -4 2 1 2 1 0 0 0| 15]\n", "[ s2| 3 1 -1 2 1 0 1 0 0| 12]\n", "[ s3| 2 0 0 2 3 0 0 1 0| 10]\n", "[ s4| 3 -1 -1 -1 1 0 0 0 1| 11]\n", "[---+-----------------------------------+---]\n", "[ Z| 3 -2 1 2 3 0 0 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tipo ='max' # Problema de tipo max o min\n", "\n", "n=5 # número de variables x's\n", "m=4 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[3,-2,1,2,3] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[1,-4,2,1,2],\n", "[3,1,-1,2,1],\n", "[2,0,0,2,3],\n", "[3,-1,-1,-1,1]\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=(6,7,8,9) # 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": 42, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -\\frac{11}{3} & \\frac{7}{3} & \\frac{4}{3} & \\frac{5}{3} & 1 & 0 & 0 & -\\frac{1}{3} & \\frac{34}{3} \\\\\n", "{\\color{green}{s}}_{2} & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", "{\\color{green}{s}}_{3} & 0 & \\frac{2}{3} & \\frac{2}{3} & \\frac{8}{3} & \\frac{7}{3} & 0 & 0 & 1 & -\\frac{2}{3} & \\frac{8}{3} \\\\\n", "{x}_{1} & 1 & -\\frac{1}{3} & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{3} & 0 & 0 & 0 & \\frac{1}{3} & \\frac{11}{3} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -1 & 2 & 3 & 2 & 0 & 0 & 0 & -1 & 11\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -\\frac{11}{3} & \\frac{7}{3} & \\frac{4}{3} & \\frac{5}{3} & 1 & 0 & 0 & -\\frac{1}{3} & \\frac{34}{3} \\\\\n", "{\\color{green}{s}}_{2} & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", "{\\color{green}{s}}_{3} & 0 & \\frac{2}{3} & \\frac{2}{3} & \\frac{8}{3} & \\frac{7}{3} & 0 & 0 & 1 & -\\frac{2}{3} & \\frac{8}{3} \\\\\n", "{x}_{1} & 1 & -\\frac{1}{3} & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{3} & 0 & 0 & 0 & \\frac{1}{3} & \\frac{11}{3} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -1 & 2 & 3 & 2 & 0 & 0 & 0 & -1 & 11\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n", "[-----+-----------------------------------------------------+-----]\n", "[ s1| 0 -11/3 7/3 4/3 5/3 1 0 0 -1/3| 34/3]\n", "[ s2| 0 2 0 3 0 0 1 0 -1| 1]\n", "[ s3| 0 2/3 2/3 8/3 7/3 0 0 1 -2/3| 8/3]\n", "[ x1| 1 -1/3 -1/3 -1/3 1/3 0 0 0 1/3| 11/3]\n", "[-----+-----------------------------------------------------+-----]\n", "[ Z| 0 -1 2 3 2 0 0 0 -1| 11]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(4,1)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & 0 & \\frac{7}{3} & \\frac{41}{6} & \\frac{5}{3} & 1 & \\frac{11}{6} & 0 & -\\frac{13}{6} & \\frac{79}{6} \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{1}{2} \\\\\n", "{\\color{green}{s}}_{3} & 0 & 0 & \\frac{2}{3} & \\frac{5}{3} & \\frac{7}{3} & 0 & -\\frac{1}{3} & 1 & -\\frac{1}{3} & \\frac{7}{3} \\\\\n", "{x}_{1} & 1 & 0 & -\\frac{1}{3} & \\frac{1}{6} & \\frac{1}{3} & 0 & \\frac{1}{6} & 0 & \\frac{1}{6} & \\frac{23}{6} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 2 & \\frac{9}{2} & 2 & 0 & \\frac{1}{2} & 0 & -\\frac{3}{2} & \\frac{21}{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & 0 & \\frac{7}{3} & \\frac{41}{6} & \\frac{5}{3} & 1 & \\frac{11}{6} & 0 & -\\frac{13}{6} & \\frac{79}{6} \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{1}{2} \\\\\n", "{\\color{green}{s}}_{3} & 0 & 0 & \\frac{2}{3} & \\frac{5}{3} & \\frac{7}{3} & 0 & -\\frac{1}{3} & 1 & -\\frac{1}{3} & \\frac{7}{3} \\\\\n", "{x}_{1} & 1 & 0 & -\\frac{1}{3} & \\frac{1}{6} & \\frac{1}{3} & 0 & \\frac{1}{6} & 0 & \\frac{1}{6} & \\frac{23}{6} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 2 & \\frac{9}{2} & 2 & 0 & \\frac{1}{2} & 0 & -\\frac{3}{2} & \\frac{21}{2}\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n", "[-----+-----------------------------------------------------+-----]\n", "[ s1| 0 0 7/3 41/6 5/3 1 11/6 0 -13/6| 79/6]\n", "[ x2| 0 1 0 3/2 0 0 1/2 0 -1/2| 1/2]\n", "[ s3| 0 0 2/3 5/3 7/3 0 -1/3 1 -1/3| 7/3]\n", "[ x1| 1 0 -1/3 1/6 1/3 0 1/6 0 1/6| 23/6]\n", "[-----+-----------------------------------------------------+-----]\n", "[ Z| 0 0 2 9/2 2 0 1/2 0 -3/2| 21/2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(2,2)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[5.6429]\n", "[ NO]\n", "[3.5000]\n", "[ NO]" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cocientes(3)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & 0 & 0 & 1 & -\\frac{13}{2} & 1 & 3 & -\\frac{7}{2} & -1 & 5 \\\\\n", "{x}_{2} & 0 & 1 & 0 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{1}{2} \\\\\n", "{x}_{3} & 0 & 0 & 1 & \\frac{5}{2} & \\frac{7}{2} & 0 & -\\frac{1}{2} & \\frac{3}{2} & -\\frac{1}{2} & \\frac{7}{2} \\\\\n", "{x}_{1} & 1 & 0 & 0 & 1 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 0 & -\\frac{1}{2} & -5 & 0 & \\frac{3}{2} & -3 & -\\frac{1}{2} & \\frac{35}{2}\n", "\\end{array}\\right)$ " ], "text/latex": [ 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"[ x2| 0 1 0 3/2 0 0 1/2 0 -1/2| 1/2]\n", "[ x3| 0 0 1 5/2 7/2 0 -1/2 3/2 -1/2| 7/2]\n", "[ x1| 1 0 0 1 3/2 0 0 1/2 0| 5]\n", "[-----+-----------------------------------------------------+-----]\n", "[ Z| 0 0 0 -1/2 -5 0 3/2 -3 -1/2| 35/2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(3,3)" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -\\frac{2}{3} & 0 & 0 & -\\frac{13}{2} & 1 & \\frac{8}{3} & -\\frac{7}{2} & -\\frac{2}{3} & \\frac{14}{3} \\\\\n", "{x}_{4} & 0 & \\frac{2}{3} & 0 & 1 & 0 & 0 & \\frac{1}{3} & 0 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n", "{x}_{3} & 0 & -\\frac{5}{3} & 1 & 0 & 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" \n" ] } ], "source": [ "tipo ='max' # Problema de tipo max o min\n", "\n", "n=5 # número de variables x's\n", "m=4 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[3,-2,1,2,3] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[1,-4,2,1,2],\n", "[3,1,-1,2,1],\n", "[2,0,0,2,3],\n", "[3,-1,-1,-1,1]\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=(6,7,8,9) # posición de la matriz identidad\n", "\n", "\n", " \n", " \n", "####################################################################\n", "############ HASTA AQUÍ 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{\\color{green}{s}}_{4} \\\\\n", "\\hline\n", " 0 & -3 & 0 & -5 & -5 & 0 & 0 & -3 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n", "{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n", "\\hline\n", " 0 & -3 & 0 & -5 & -5 & 0 & 0 & -3 & 1\n", "\\end{array}\\right)$$" ], "text/plain": [ "[x1 x2 x3 x4 x5 s1 s2 s3 s4]\n", "[--------------------------]\n", "[ 0 -3 0 -5 -5 0 0 -3 1]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ " \n" ] } ], "source": [ "cambio(2,7)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr|rr}\n", "{Basic} & {b} & {Col\\ Entr} & {Cocientes} \\\\\n", "{\\color{green}{s}}_{1} & 2 & 2 & 1.0000 \\\\\n", 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"\\end{array}\\right)$$" ], "text/plain": [ "[ B| s4 s2 x3 x1| rhs]\n", "[----+-------------------+----]\n", "[ s4| 1/2 0 -7/4 1| 1]\n", "[ s2| 1/2 1 -7/4 0| 2]\n", "[ x3| 1/2 0 -1/4 0| 5]\n", "[ x1| 0 0 1/2 0| 5]\n", "[----+-------------------+----]\n", "[ ZPI| 1/2 0 5/4 0| 20]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Fila Indicadora\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n", "{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n", "\\hline\n", " 0 & 0 & 0 & -1 & -\\frac{7}{4} & -\\frac{1}{2} & 0 & -\\frac{5}{4} & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n", "{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & 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\\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 x_{5} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n", " \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 3 x_{1} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} - \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 11 \\\\\n", "\\end{array} \\\\\n", "x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \\geq 0\n", "\\end{array}$$" ], "text/plain": [ "LP problem (use 'view(...)' or '%display typeset' for details)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}20$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}20$$" ], "text/plain": [ "20" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(5,\\,0,\\,5,\\,0,\\,0\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(5,\\,0,\\,5,\\,0,\\,0\\right)$$" ], "text/plain": [ "(5, 0, 5, 0, 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": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 1 & -4 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & 15 \\\\\n", "{\\color{green}{s}}_{2} & 3 & 1 & -1 & 2 & 1 & 0 & 1 & 0 & 0 & 12 \\\\\n", "{\\color{green}{s}}_{3} & 2 & 0 & 0 & 2 & 3 & 0 & 0 & 1 & 0 & 10 \\\\\n", "{\\color{green}{s}}_{4} & 3 & -1 & -1 & -1 & 1 & 0 & 0 & 0 & 1 & 11 \\\\\n", "\\hline\n", " {Z_{ind}} & 3 & -2 & 1 & 2 & 3 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 1 & -4 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & 15 \\\\\n", "{\\color{green}{s}}_{2} & 3 & 1 & -1 & 2 & 1 & 0 & 1 & 0 & 0 & 12 \\\\\n", "{\\color{green}{s}}_{3} & 2 & 0 & 0 & 2 & 3 & 0 & 0 & 1 & 0 & 10 \\\\\n", "{\\color{green}{s}}_{4} & 3 & -1 & -1 & -1 & 1 & 0 & 0 & 0 & 1 & 11 \\\\\n", "\\hline\n", " {Z_{ind}} & 3 & -2 & 1 & 2 & 3 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4|rhs]\n", "[---+-----------------------------------+---]\n", "[ s1| 1 -4 2 1 2 1 0 0 0| 15]\n", "[ s2| 3 1 -1 2 1 0 1 0 0| 12]\n", "[ s3| 2 0 0 2 3 0 0 1 0| 10]\n", "[ s4| 3 -1 -1 -1 1 0 0 0 1| 11]\n", "[---+-----------------------------------+---]\n", "[ Z| 3 -2 1 2 3 0 0 0 0| 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tipo ='max' # Problema de tipo max o min\n", "\n", "n=5 # número de variables x's\n", "m=4 # número de restricciones (distintas de las de signo)\n", "\n", "c=matrix(QQ,1,n,[3,-2,1,2,3] ) # vector de costos\n", "\n", "A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n", "[1,-4,2,1,2],\n", "[3,1,-1,2,1],\n", "[2,0,0,2,3],\n", "[3,-1,-1,-1,1]\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=(6,7,8,9) # 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": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -\\frac{11}{3} & \\frac{7}{3} & \\frac{4}{3} & \\frac{5}{3} & 1 & 0 & 0 & -\\frac{1}{3} & \\frac{34}{3} \\\\\n", "{\\color{green}{s}}_{2} & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", "{\\color{green}{s}}_{3} & 0 & \\frac{2}{3} & \\frac{2}{3} & \\frac{8}{3} & \\frac{7}{3} & 0 & 0 & 1 & -\\frac{2}{3} & \\frac{8}{3} \\\\\n", "{x}_{1} & 1 & -\\frac{1}{3} & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{3} & 0 & 0 & 0 & \\frac{1}{3} & \\frac{11}{3} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -1 & 2 & 3 & 2 & 0 & 0 & 0 & -1 & 11\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -\\frac{11}{3} & \\frac{7}{3} & \\frac{4}{3} & \\frac{5}{3} & 1 & 0 & 0 & -\\frac{1}{3} & \\frac{34}{3} \\\\\n", "{\\color{green}{s}}_{2} & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", "{\\color{green}{s}}_{3} & 0 & \\frac{2}{3} & \\frac{2}{3} & \\frac{8}{3} & \\frac{7}{3} & 0 & 0 & 1 & -\\frac{2}{3} & \\frac{8}{3} \\\\\n", "{x}_{1} & 1 & -\\frac{1}{3} & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{3} & 0 & 0 & 0 & \\frac{1}{3} & \\frac{11}{3} \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -1 & 2 & 3 & 2 & 0 & 0 & 0 & -1 & 11\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n", "[-----+-----------------------------------------------------+-----]\n", "[ s1| 0 -11/3 7/3 4/3 5/3 1 0 0 -1/3| 34/3]\n", "[ s2| 0 2 0 3 0 0 1 0 -1| 1]\n", "[ s3| 0 2/3 2/3 8/3 7/3 0 0 1 -2/3| 8/3]\n", "[ x1| 1 -1/3 -1/3 -1/3 1/3 0 0 0 1/3| 11/3]\n", "[-----+-----------------------------------------------------+-----]\n", "[ Z| 0 -1 2 3 2 0 0 0 -1| 11]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -6 & 0 & -8 & -\\frac{13}{2} & 1 & 0 & -\\frac{7}{2} & 2 & 2 \\\\\n", "{\\color{green}{s}}_{2} & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", "{x}_{3} & 0 & 1 & 1 & 4 & \\frac{7}{2} & 0 & 0 & \\frac{3}{2} & -1 & 4 \\\\\n", "{x}_{1} & 1 & 0 & 0 & 1 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -3 & 0 & -5 & -5 & 0 & 0 & -3 & 1 & 19\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{1} & 0 & -6 & 0 & -8 & -\\frac{13}{2} & 1 & 0 & -\\frac{7}{2} & 2 & 2 \\\\\n", "{\\color{green}{s}}_{2} & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 0 & -1 & 1 \\\\\n", "{x}_{3} & 0 & 1 & 1 & 4 & \\frac{7}{2} & 0 & 0 & \\frac{3}{2} & -1 & 4 \\\\\n", "{x}_{1} & 1 & 0 & 0 & 1 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & -3 & 0 & -5 & -5 & 0 & 0 & -3 & 1 & 19\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n", "[-----+-----------------------------------------------------+-----]\n", "[ s1| 0 -6 0 -8 -13/2 1 0 -7/2 2| 2]\n", "[ s2| 0 2 0 3 0 0 1 0 -1| 1]\n", "[ x3| 0 1 1 4 7/2 0 0 3/2 -1| 4]\n", "[ x1| 1 0 0 1 3/2 0 0 1/2 0| 5]\n", "[-----+-----------------------------------------------------+-----]\n", "[ Z| 0 -3 0 -5 -5 0 0 -3 1| 19]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{4} & 0 & -3 & 0 & -4 & -\\frac{13}{4} & \\frac{1}{2} & 0 & -\\frac{7}{4} & 1 & 1 \\\\\n", "{\\color{green}{s}}_{2} & 0 & -1 & 0 & -1 & -\\frac{13}{4} & \\frac{1}{2} & 1 & -\\frac{7}{4} & 0 & 2 \\\\\n", "{x}_{3} & 0 & -2 & 1 & 0 & \\frac{1}{4} & \\frac{1}{2} & 0 & -\\frac{1}{4} & 0 & 5 \\\\\n", "{x}_{1} & 1 & 0 & 0 & 1 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 0 & -1 & -\\frac{7}{4} & -\\frac{1}{2} & 0 & -\\frac{5}{4} & 0 & 20\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n", "{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n", "\\hline\n", " {\\color{green}{s}}_{4} & 0 & -3 & 0 & -4 & -\\frac{13}{4} & \\frac{1}{2} & 0 & -\\frac{7}{4} & 1 & 1 \\\\\n", "{\\color{green}{s}}_{2} & 0 & -1 & 0 & -1 & -\\frac{13}{4} & \\frac{1}{2} & 1 & -\\frac{7}{4} & 0 & 2 \\\\\n", "{x}_{3} & 0 & -2 & 1 & 0 & \\frac{1}{4} & \\frac{1}{2} & 0 & -\\frac{1}{4} & 0 & 5 \\\\\n", "{x}_{1} & 1 & 0 & 0 & 1 & \\frac{3}{2} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n", "\\hline\n", " {Z_{ind}} & 0 & 0 & 0 & -1 & -\\frac{7}{4} & -\\frac{1}{2} & 0 & -\\frac{5}{4} & 0 & 20\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n", "[-----+-----------------------------------------------------+-----]\n", "[ s4| 0 -3 0 -4 -13/4 1/2 0 -7/4 1| 1]\n", "[ s2| 0 -1 0 -1 -13/4 1/2 1 -7/4 0| 2]\n", "[ x3| 0 -2 1 0 1/4 1/2 0 -1/4 0| 5]\n", "[ x1| 1 0 0 1 3/2 0 0 1/2 0| 5]\n", "[-----+-----------------------------------------------------+-----]\n", "[ Z| 0 0 0 -1 -7/4 -1/2 0 -5/4 0| 20]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cambio(4,1);cambio(3,3);cambio(1,9)" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", "\\frac{1}{2} & 0 & 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