{
"cells": [
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n",
"\\begin{array}{lcrcrcrcrcl}\n",
" \\max \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 10 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 20 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 40 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 32 x_{4} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n",
" \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 10 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 30 x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 80 x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 40 x_{4} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 900 \\\\\n",
" \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 2 x_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{2} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} x_{3} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 x_{4} \\mspace{-6mu}&\\mspace{-6mu} \\leq \\mspace{-6mu}&\\mspace{-6mu} 80 \\\\\n",
"\\end{array} \\\\\n",
"x_{1}, x_{2}, x_{3}, x_{4} \\geq 0\n",
"\\end{array}$$"
],
"text/plain": [
"LP problem (use 'view(...)' or '%display typeset' for details)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}740$$"
],
"text/plain": [
"740"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(10,\\,0,\\,0,\\,20\\right)$$"
],
"text/plain": [
"(10, 0, 0, 20)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$"
],
"text/plain": [
"salto"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\texttt{El dual es:}}$$"
],
"text/plain": [
"dual"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{l}\n",
"\\begin{array}{lcrcrcl}\n",
" \\min \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 900 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 80 y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} \\\\\n",
" \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 10 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 2 y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 10 \\\\\n",
" \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 30 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 20 \\\\\n",
" \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 80 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 40 \\\\\n",
" \\mspace{-6mu}&\\mspace{-6mu} \\mspace{-6mu}&\\mspace{-6mu} 40 y_{1} \\mspace{-6mu}&\\mspace{-6mu} + \\mspace{-6mu}&\\mspace{-6mu} 3 y_{2} \\mspace{-6mu}&\\mspace{-6mu} \\geq \\mspace{-6mu}&\\mspace{-6mu} 32 \\\\\n",
"\\end{array} \\\\\n",
"y_{1}, y_{2} \\geq 0\n",
"\\end{array}$$"
],
"text/plain": [
"LP problem (use 'view(...)' or '%display typeset' for details)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}740$$"
],
"text/plain": [
"740"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{17}{25},\\,\\frac{8}{5}\\right)$$"
],
"text/plain": [
"(17/25, 8/5)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{}$$"
],
"text/plain": [
"salto"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{===================================}$$"
],
"text/plain": [
"ry"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n",
"{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n",
"\\hline\n",
" {\\color{green}{s}}_{1} & 10 & 30 & 80 & 40 & 1 & 0 & 900 \\\\\n",
"{\\color{green}{s}}_{2} & 2 & 1 & 1 & 3 & 0 & 1 & 80 \\\\\n",
"\\hline\n",
" {Z_{ind}} & 10 & 20 & 40 & 32 & 0 & 0 & 0\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| x1 x2 x3 x4 s1 s2|rhs]\n",
"[---+-----------------------+---]\n",
"[ s1| 10 30 80 40 1 0|900]\n",
"[ s2| 2 1 1 3 0 1| 80]\n",
"[---+-----------------------+---]\n",
"[ Z| 10 20 40 32 0 0| 0]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"cambio( 1 , 3 )\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n",
"{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n",
"\\hline\n",
" {x}_{3} & \\frac{1}{8} & \\frac{3}{8} & 1 & \\frac{1}{2} & \\frac{1}{80} & 0 & \\frac{45}{4} \\\\\n",
"{\\color{green}{s}}_{2} & \\frac{15}{8} & \\frac{5}{8} & 0 & \\frac{5}{2} & -\\frac{1}{80} & 1 & \\frac{275}{4} \\\\\n",
"\\hline\n",
" {Z_{ind}} & 5 & 5 & 0 & 12 & -\\frac{1}{2} & 0 & 450\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| x1 x2 x3 x4 s1 s2| rhs]\n",
"[-----+-----------------------------------+-----]\n",
"[ x3| 1/8 3/8 1 1/2 1/80 0| 45/4]\n",
"[ s2| 15/8 5/8 0 5/2 -1/80 1|275/4]\n",
"[-----+-----------------------------------+-----]\n",
"[ Z| 5 5 0 12 -1/2 0| 450]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"cambio( 1 , 4 )\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n",
"{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n",
"\\hline\n",
" {x}_{4} & \\frac{1}{4} & \\frac{3}{4} & 2 & 1 & \\frac{1}{40} & 0 & \\frac{45}{2} \\\\\n",
"{\\color{green}{s}}_{2} & \\frac{5}{4} & -\\frac{5}{4} & -5 & 0 & -\\frac{3}{40} & 1 & \\frac{25}{2} \\\\\n",
"\\hline\n",
" {Z_{ind}} & 2 & -4 & -24 & 0 & -\\frac{4}{5} & 0 & 720\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| x1 x2 x3 x4 s1 s2| rhs]\n",
"[-----+-----------------------------------+-----]\n",
"[ x4| 1/4 3/4 2 1 1/40 0| 45/2]\n",
"[ s2| 5/4 -5/4 -5 0 -3/40 1| 25/2]\n",
"[-----+-----------------------------------+-----]\n",
"[ Z| 2 -4 -24 0 -4/5 0| 720]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"cambio( 2 , 1 )\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r|rrrrrr|r}\n",
"{Basic} & {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} & {\\color{green}{s}}_{1} & {\\color{green}{s}}_{2} & {b} \\\\\n",
"\\hline\n",
" {x}_{4} & 0 & 1 & 3 & 1 & \\frac{1}{25} & -\\frac{1}{5} & 20 \\\\\n",
"{x}_{1} & 1 & -1 & -4 & 0 & -\\frac{3}{50} & \\frac{4}{5} & 10 \\\\\n",
"\\hline\n",
" {Z_{ind}} & 0 & -2 & -16 & 0 & -\\frac{17}{25} & -\\frac{8}{5} & 740\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ B| x1 x2 x3 x4 s1 s2| rhs]\n",
"[------+-----------------------------------------+------]\n",
"[ x4| 0 1 3 1 1/25 -1/5| 20]\n",
"[ x1| 1 -1 -4 0 -3/50 4/5| 10]\n",
"[------+-----------------------------------------+------]\n",
"[ Z| 0 -2 -16 0 -17/25 -8/5| 740]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"El problema es óptimo\n"
]
}
],
"source": [
"tipo='max' # max o min\n",
"\n",
"n=4 # número de variables x's\n",
"m=2 # número de restricciones (distintas de las de signo)\n",
"\n",
"coste=matrix(QQ,1,n,[20,30,45,58] )\n",
"venta=matrix(QQ,1,n,[30,50,85,90] )\n",
"\n",
"c=venta-coste # vector de costos\n",
"\n",
"A=matrix(QQ,m,n,[ # coeficientes de las variables x's\n",
"[10,30,80,40],\n",
"[2,1,1,3]\n",
"])\n",
"\n",
"b=matrix(QQ,m,1,[900,80]) # vector de la derecha\n",
"\n",
"\n",
"h=matrix(ZZ,1,m,[1,1]) # variables de holgura: \n",
" # Pon 1 si va sumando, \n",
" # -1 si va restando y \n",
" # 0 si no hay variable de holgura.\n",
"#######################################################\n",
"\n",
"\n",
"load('https://sage.unex.es/501708/simplex2023Auto.sage')\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
"\\frac{1}{25} & -\\frac{1}{5} \\\\\n",
"-\\frac{3}{50} & \\frac{4}{5}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ 1/25 -1/5]\n",
"[-3/50 4/5]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(BM1)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.5",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.2"
}
},
"nbformat": 4,
"nbformat_minor": 4
}