{
"cells": [
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/usr/lib/python3/dist-packages/sage/misc/remote_file.py:46: DeprecationWarning: ssl.SSLContext() without protocol argument is deprecated.\n",
" content = urlopen(req, timeout=1, context=SSLContext())\n",
"/usr/lib/python3/dist-packages/sage/misc/remote_file.py:46: DeprecationWarning: ssl.PROTOCOL_TLS is deprecated\n",
" content = urlopen(req, timeout=1, context=SSLContext())\n"
]
},
{
"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}{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} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\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} \\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} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{5} \\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} - \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\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} \\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": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}20$$"
],
"text/plain": [
"20"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"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": [
""
],
"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": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"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')\n",
"#load('https://sage.unex.es/501708/simplex2023Auto.sage')"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[15.000]\n",
"[4.0000]\n",
"[5.0000]\n",
"[3.6667]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cocientes(1)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"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": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[ 8.5000]\n",
"[0.33333]\n",
"[ 1.0000]\n",
"[ NO]"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cocientes(4)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n",
"{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 0 & -\\frac{41}{9} & \\frac{7}{3} & 0 & \\frac{5}{3} & 1 & -\\frac{4}{9} & 0 & \\frac{1}{9} & \\frac{98}{9} \\\\\n",
"{x}_{4} & 0 & \\frac{2}{3} & 0 & 1 & 0 & 0 & \\frac{1}{3} & 0 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
"{\\color{green}{s}}_{3} & 0 & -\\frac{10}{9} & \\frac{2}{3} & 0 & \\frac{7}{3} & 0 & -\\frac{8}{9} & 1 & \\frac{2}{9} & \\frac{16}{9} \\\\\n",
"{x}_{1} & 1 & -\\frac{1}{9} & -\\frac{1}{3} & 0 & \\frac{1}{3} & 0 & \\frac{1}{9} & 0 & \\frac{2}{9} & \\frac{34}{9} \\\\\n",
"\\hline\n",
" {Z_{ind}} & 0 & -3 & 2 & 0 & 2 & 0 & -1 & 0 & 0 & 12\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n",
"[-----+-----------------------------------------------------+-----]\n",
"[ s1| 0 -41/9 7/3 0 5/3 1 -4/9 0 1/9| 98/9]\n",
"[ x4| 0 2/3 0 1 0 0 1/3 0 -1/3| 1/3]\n",
"[ s3| 0 -10/9 2/3 0 7/3 0 -8/9 1 2/9| 16/9]\n",
"[ x1| 1 -1/9 -1/3 0 1/3 0 1/9 0 2/9| 34/9]\n",
"[-----+-----------------------------------------------------+-----]\n",
"[ Z| 0 -3 2 0 2 0 -1 0 0| 12]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"cambio(2,4)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[4.6667]\n",
"[ NO]\n",
"[2.6667]\n",
"[ NO]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cocientes(3)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrrrrr|r}\n",
"{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 0 & -\\frac{2}{3} & 0 & 0 & -\\frac{13}{2} & 1 & \\frac{8}{3} & -\\frac{7}{2} & -\\frac{2}{3} & \\frac{14}{3} \\\\\n",
"{x}_{4} & 0 & \\frac{2}{3} & 0 & 1 & 0 & 0 & \\frac{1}{3} & 0 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
"{x}_{3} & 0 & -\\frac{5}{3} & 1 & 0 & \\frac{7}{2} & 0 & -\\frac{4}{3} & \\frac{3}{2} & \\frac{1}{3} & \\frac{8}{3} \\\\\n",
"{x}_{1} & 1 & -\\frac{2}{3} & 0 & 0 & \\frac{3}{2} & 0 & -\\frac{1}{3} & \\frac{1}{2} & \\frac{1}{3} & \\frac{14}{3} \\\\\n",
"\\hline\n",
" {Z_{ind}} & 0 & \\frac{1}{3} & 0 & 0 & -5 & 0 & \\frac{5}{3} & -3 & -\\frac{2}{3} & \\frac{52}{3}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| x1 x2 x3 x4 x5 s1 s2 s3 s4| rhs]\n",
"[-----+-----------------------------------------------------+-----]\n",
"[ s1| 0 -2/3 0 0 -13/2 1 8/3 -7/2 -2/3| 14/3]\n",
"[ x4| 0 2/3 0 1 0 0 1/3 0 -1/3| 1/3]\n",
"[ x3| 0 -5/3 1 0 7/2 0 -4/3 3/2 1/3| 8/3]\n",
"[ x1| 1 -2/3 0 0 3/2 0 -1/3 1/2 1/3| 14/3]\n",
"[-----+-----------------------------------------------------+-----]\n",
"[ Z| 0 1/3 0 0 -5 0 5/3 -3 -2/3| 52/3]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"cambio(3,3)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"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"
}
],
"source": [
"cambio(2,7)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"cambio(1,9)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
"1 & 0 & -\\frac{7}{2} & 2 \\\\\n",
"0 & 1 & 0 & -1 \\\\\n",
"0 & 0 & \\frac{3}{2} & -1 \\\\\n",
"0 & 0 & \\frac{1}{2} & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ 1 0 -7/2 2]\n",
"[ 0 1 0 -1]\n",
"[ 0 0 3/2 -1]\n",
"[ 0 0 1/2 0]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(BM1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Otro "
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"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}{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} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\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} \\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} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{5} \\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} - \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\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} \\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": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}20$$"
],
"text/plain": [
"20"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"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": [
""
],
"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": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"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"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n",
"{Basic} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 1 & 0 & 0 & 0 & 15 \\\\\n",
"{\\color{green}{s}}_{2} & 0 & 1 & 0 & 0 & 12 \\\\\n",
"{\\color{green}{s}}_{3} & 0 & 0 & 1 & 0 & 10 \\\\\n",
"{\\color{green}{s}}_{4} & 0 & 0 & 0 & 1 & 11 \\\\\n",
"\\hline\n",
" {\\pi_B} & 0 & 0 & 0 & 0 & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| s1 s2 s3 s4|rhs]\n",
"[---+---------------+---]\n",
"[ s1| 1 0 0 0| 15]\n",
"[ s2| 0 1 0 0| 12]\n",
"[ s3| 0 0 1 0| 10]\n",
"[ s4| 0 0 0 1| 11]\n",
"[---+---------------+---]\n",
"[ZPI| 0 0 0 0| 0]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Fila Indicadora\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n",
"{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n",
"\\hline\n",
" 3 & -2 & 1 & 2 & 3 & 0 & 0 & 0 & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[x1 x2 x3 x4 x5 s1 s2 s3 s4]\n",
"[--------------------------]\n",
"[ 3 -2 1 2 3 0 0 0 0]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" \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 LOS DATOS ################################\n",
"####################################################################\n",
"\n",
"\n",
"\n",
"\n",
"load('https://sage.unex.es/501708/simplex2023ManualRevisada.sage')"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n",
" Matriz Inversa\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n",
"{Basic} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {x}_{1} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 1 & 0 & 0 & -\\frac{1}{3} & \\frac{34}{3} \\\\\n",
"{\\color{green}{s}}_{2} & 0 & 1 & 0 & -1 & 1 \\\\\n",
"{\\color{green}{s}}_{3} & 0 & 0 & 1 & -\\frac{2}{3} & \\frac{8}{3} \\\\\n",
"{x}_{1} & 0 & 0 & 0 & \\frac{1}{3} & \\frac{11}{3} \\\\\n",
"\\hline\n",
" {\\pi_B} & 0 & 0 & 0 & 1 & 11\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| s1 s2 s3 x1| rhs]\n",
"[----+-------------------+----]\n",
"[ s1| 1 0 0 -1/3|34/3]\n",
"[ s2| 0 1 0 -1| 1]\n",
"[ s3| 0 0 1 -2/3| 8/3]\n",
"[ x1| 0 0 0 1/3|11/3]\n",
"[----+-------------------+----]\n",
"[ ZPI| 0 0 0 1| 11]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Fila Indicadora\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n",
"{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n",
"\\hline\n",
" 0 & -1 & 2 & 3 & 2 & 0 & 0 & 0 & -1\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[x1 x2 x3 x4 x5 s1 s2 s3 s4]\n",
"[--------------------------]\n",
"[ 0 -1 2 3 2 0 0 0 -1]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n"
]
}
],
"source": [
"cambio(4,1)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr|rr}\n",
"{Basic} & {b} & {Col\\ Entr} & {Cocientes} \\\\\n",
"{\\color{green}{s}}_{1} & \\frac{34}{3} & \\frac{4}{3} & 8.5000 \\\\\n",
"{\\color{green}{s}}_{2} & 1 & 3 & 0.33333 \\\\\n",
"{\\color{green}{s}}_{3} & \\frac{8}{3} & \\frac{8}{3} & 1.0000 \\\\\n",
"{x}_{1} & \\frac{11}{3} & -\\frac{1}{3} & \\mathit{NO}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B rhs| Ent Coc]\n",
"[ s1 34/3| 4/3 8.5000]\n",
"[ s2 1| 3 0.33333]\n",
"[ s3 8/3| 8/3 1.0000]\n",
"[ x1 11/3| -1/3 NO]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"cocientes(4)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n",
" Matriz Inversa\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n",
"{Basic} & {\\color{green}{s}}_{1} & {x}_{4} & {\\color{green}{s}}_{3} & {x}_{1} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 1 & -\\frac{4}{9} & 0 & \\frac{1}{9} & \\frac{98}{9} \\\\\n",
"{x}_{4} & 0 & \\frac{1}{3} & 0 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
"{\\color{green}{s}}_{3} & 0 & -\\frac{8}{9} & 1 & \\frac{2}{9} & \\frac{16}{9} \\\\\n",
"{x}_{1} & 0 & \\frac{1}{9} & 0 & \\frac{2}{9} & \\frac{34}{9} \\\\\n",
"\\hline\n",
" {\\pi_B} & 0 & 1 & 0 & 0 & 12\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| s1 x4 s3 x1| rhs]\n",
"[----+-------------------+----]\n",
"[ s1| 1 -4/9 0 1/9|98/9]\n",
"[ x4| 0 1/3 0 -1/3| 1/3]\n",
"[ s3| 0 -8/9 1 2/9|16/9]\n",
"[ x1| 0 1/9 0 2/9|34/9]\n",
"[----+-------------------+----]\n",
"[ ZPI| 0 1 0 0| 12]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Fila Indicadora\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n",
"{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n",
"\\hline\n",
" 0 & -3 & 2 & 0 & 2 & 0 & -1 & 0 & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[x1 x2 x3 x4 x5 s1 s2 s3 s4]\n",
"[--------------------------]\n",
"[ 0 -3 2 0 2 0 -1 0 0]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n"
]
}
],
"source": [
"cambio(2,4)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr|rr}\n",
"{Basic} & {b} & {Col\\ Entr} & {Cocientes} \\\\\n",
"{\\color{green}{s}}_{1} & \\frac{98}{9} & \\frac{7}{3} & 4.6667 \\\\\n",
"{x}_{4} & \\frac{1}{3} & 0 & \\mathit{NO} \\\\\n",
"{\\color{green}{s}}_{3} & \\frac{16}{9} & \\frac{2}{3} & 2.6667 \\\\\n",
"{x}_{1} & \\frac{34}{9} & -\\frac{1}{3} & \\mathit{NO}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B rhs| Ent Coc]\n",
"[ s1 98/9| 7/3 4.6667]\n",
"[ x4 1/3| 0 NO]\n",
"[ s3 16/9| 2/3 2.6667]\n",
"[ x1 34/9| -1/3 NO]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"cocientes(3)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n",
" Matriz Inversa\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n",
"{Basic} & {\\color{green}{s}}_{1} & {x}_{4} & {x}_{3} & {x}_{1} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 1 & \\frac{8}{3} & -\\frac{7}{2} & -\\frac{2}{3} & \\frac{14}{3} \\\\\n",
"{x}_{4} & 0 & \\frac{1}{3} & 0 & -\\frac{1}{3} & \\frac{1}{3} \\\\\n",
"{x}_{3} & 0 & -\\frac{4}{3} & \\frac{3}{2} & \\frac{1}{3} & \\frac{8}{3} \\\\\n",
"{x}_{1} & 0 & -\\frac{1}{3} & \\frac{1}{2} & \\frac{1}{3} & \\frac{14}{3} \\\\\n",
"\\hline\n",
" {\\pi_B} & 0 & -\\frac{5}{3} & 3 & \\frac{2}{3} & \\frac{52}{3}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| s1 x4 x3 x1| rhs]\n",
"[----+-------------------+----]\n",
"[ s1| 1 8/3 -7/2 -2/3|14/3]\n",
"[ x4| 0 1/3 0 -1/3| 1/3]\n",
"[ x3| 0 -4/3 3/2 1/3| 8/3]\n",
"[ x1| 0 -1/3 1/2 1/3|14/3]\n",
"[----+-------------------+----]\n",
"[ ZPI| 0 -5/3 3 2/3|52/3]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Fila Indicadora\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n",
"{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n",
"\\hline\n",
" 0 & \\frac{1}{3} & 0 & 0 & -5 & 0 & \\frac{5}{3} & -3 & -\\frac{2}{3}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ x1 x2 x3 x4 x5 s1 s2 s3 s4]\n",
"[--------------------------------------------]\n",
"[ 0 1/3 0 0 -5 0 5/3 -3 -2/3]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n"
]
}
],
"source": [
"cambio(3,3)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr|rr}\n",
"{Basic} & {b} & {Col\\ Entr} & {Cocientes} \\\\\n",
"{\\color{green}{s}}_{1} & \\frac{14}{3} & \\frac{8}{3} & 1.7500 \\\\\n",
"{x}_{4} & \\frac{1}{3} & \\frac{1}{3} & 1.0000 \\\\\n",
"{x}_{3} & \\frac{8}{3} & -\\frac{4}{3} & \\mathit{NO} \\\\\n",
"{x}_{1} & \\frac{14}{3} & -\\frac{1}{3} & \\mathit{NO}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B rhs| Ent Coc]\n",
"[ s1 14/3| 8/3 1.7500]\n",
"[ x4 1/3| 1/3 1.0000]\n",
"[ x3 8/3| -4/3 NO]\n",
"[ x1 14/3| -1/3 NO]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"cocientes(7)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n",
" Matriz Inversa\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n",
"{Basic} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {x}_{3} & {x}_{1} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 1 & 0 & -\\frac{7}{2} & 2 & 2 \\\\\n",
"{\\color{green}{s}}_{2} & 0 & 1 & 0 & -1 & 1 \\\\\n",
"{x}_{3} & 0 & 0 & \\frac{3}{2} & -1 & 4 \\\\\n",
"{x}_{1} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n",
"\\hline\n",
" {\\pi_B} & 0 & 0 & 3 & -1 & 19\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| s1 s2 x3 x1| rhs]\n",
"[----+-------------------+----]\n",
"[ s1| 1 0 -7/2 2| 2]\n",
"[ s2| 0 1 0 -1| 1]\n",
"[ x3| 0 0 3/2 -1| 4]\n",
"[ x1| 0 0 1/2 0| 5]\n",
"[----+-------------------+----]\n",
"[ ZPI| 0 0 3 -1| 19]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Fila Indicadora\n"
]
},
{
"data": {
"text/html": [
""
],
"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": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr|rr}\n",
"{Basic} & {b} & {Col\\ Entr} & {Cocientes} \\\\\n",
"{\\color{green}{s}}_{1} & 2 & 2 & 1.0000 \\\\\n",
"{\\color{green}{s}}_{2} & 1 & -1 & \\mathit{NO} \\\\\n",
"{x}_{3} & 4 & -1 & \\mathit{NO} \\\\\n",
"{x}_{1} & 5 & 0 & \\mathit{NO}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B rhs| Ent Coc]\n",
"[ s1 2| 2 1.0000]\n",
"[ s2 1| -1 NO]\n",
"[ x3 4| -1 NO]\n",
"[ x1 5| 0 NO]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"cocientes(9)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n",
" Matriz Inversa\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrr|r}\n",
"{Basic} & {\\color{green}{s}}_{4} & {\\color{green}{s}}_{2} & {x}_{3} & {x}_{1} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{4} & \\frac{1}{2} & 0 & -\\frac{7}{4} & 1 & 1 \\\\\n",
"{\\color{green}{s}}_{2} & \\frac{1}{2} & 1 & -\\frac{7}{4} & 0 & 2 \\\\\n",
"{x}_{3} & \\frac{1}{2} & 0 & -\\frac{1}{4} & 0 & 5 \\\\\n",
"{x}_{1} & 0 & 0 & \\frac{1}{2} & 0 & 5 \\\\\n",
"\\hline\n",
" {\\pi_B} & \\frac{1}{2} & 0 & \\frac{5}{4} & 0 & 20\n",
"\\end{array}\\right)$$"
],
"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": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrr}\n",
"{x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {x}_{5} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {\\color{green}{s}}_{3} & {\\color{green}{s}}_{4} \\\\\n",
"\\hline\n",
" 0 & 0 & 0 & -1 & -\\frac{7}{4} & -\\frac{1}{2} & 0 & -\\frac{5}{4} & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ x1 x2 x3 x4 x5 s1 s2 s3 s4]\n",
"[--------------------------------------------]\n",
"[ 0 0 0 -1 -7/4 -1/2 0 -5/4 0]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" \n"
]
}
],
"source": [
"cambio(1,9)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Otro "
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"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}{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} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\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} \\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} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{5} \\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} - \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 x_{4} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{5} \\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} \\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": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}20$$"
],
"text/plain": [
"20"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"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": [
""
],
"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": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"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": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"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": [
""
],
"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": [
""
],
"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": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
"\\frac{1}{2} & 0 & -\\frac{7}{4} & 1 \\\\\n",
"\\frac{1}{2} & 1 & -\\frac{7}{4} & 0 \\\\\n",
"\\frac{1}{2} & 0 & -\\frac{1}{4} & 0 \\\\\n",
"0 & 0 & \\frac{1}{2} & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ 1/2 0 -7/4 1]\n",
"[ 1/2 1 -7/4 0]\n",
"[ 1/2 0 -1/4 0]\n",
"[ 0 0 1/2 0]"
]
},
"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",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.2"
}
},
"nbformat": 4,
"nbformat_minor": 4
}