[13.2] Production Problem#
Revised
24 Apr 2023
Programming Environment#
Show code cell source
from fractions import Fraction
import numpy as np
import sympy as sp
from sympy import init_printing,latex,nsimplify,solve,symbols
from sympy.matrices import diag,eye,GramSchmidt,ones,Matrix,zeros
from sympy.solvers.inequalities import reduce_inequalities
x1,x2,x3,x4,y1,y2,y3,y4=symbols('x1 x2 x3 x4 y1 y2 y3 y4')
init_printing(use_latex=True)
import matplotlib as mpl
import matplotlib.pyplot as plt
plt.style.use('ggplot');
plt.rcParams.update({'text.usetex' : True});
%matplotlib inline
from IPython.display import display, Math
from datetime import datetime as d
import locale as l
import platform as p
import sys as s
pad = 20
print(f"{'Executed'.upper():<{pad}}: {d.now()}")
print()
print(f"{'Platform' :<{pad}}: "
f"{p.mac_ver()[0]} | "
f"{p.system()} | "
f"{p.release()} | "
f"{p.machine()}")
print(f"{'' :<{pad}}: {l.getpreferredencoding()}")
print()
print(f"{'Python' :<{pad}}: {s.version}")
print(f"{'' :<{pad}}: {s.version_info}")
print(f"{'' :<{pad}}: {p.python_implementation()}")
print()
print(f"{'Matplotlib' :<{pad}}: {mpl.__version__}")
print(f"{'NumPy' :<{pad}}: {np .__version__}")
print(f"{'SymPy' :<{pad}}: {sp .__version__}")
def unitVector (vec : np.ndarray) -> np.ndarray:
return vec/np.sqrt((vec[0]**2+vec[1]**2))
EXECUTED : 2024-05-21 15:46:42.168814
Platform : 14.4.1 | Darwin | 23.4.0 | arm64
: UTF-8
Python : 3.11.9 | packaged by conda-forge | (main, Apr 19 2024, 18:34:54) [Clang 16.0.6 ]
: sys.version_info(major=3, minor=11, micro=9, releaselevel='final', serial=0)
: CPython
Matplotlib : 3.8.4
NumPy : 1.26.4
SymPy : 1.12
Pivot Function#
# PIVOT FUNCTION FOR CONCISE ROW-COLUMN TABLEAU
def pivot (iStar,jStar,tab):
nRows =tab.shape[0]-1
nCols =tab.shape[1]-1
newTab=tab[:,:]
newTab[0,jStar] =tab[iStar,nCols]
newTab[iStar,nCols]=tab[0,jStar]
newTab[iStar,0] =-tab[nRows,jStar]
newTab[nRows,jStar]=-tab[iStar,0]
for i in range(1,nRows):
for j in range(1,nCols):
if i==iStar and j==jStar:
newTab[i,j]=1/tab[iStar,jStar]
elif i==iStar and j!=jStar:
newTab[i,j]=-tab[i,j]/tab[iStar,jStar]
elif i!=iStar and j==jStar:
newTab[i,j]= tab[i,j]/tab[iStar,jStar]
elif i!=iStar and j!=jStar:
newTab[i,j]= tab[i,j]-tab[iStar,j]*tab[i,jStar]/tab[iStar,jStar]
return newTab
Production Scenario#
Product A [x1] (metal valve Type A)
$500.00 sale price
20 lb material
10 hr labor
25 unit demand per week
Product B [x2] (metal valve Type B)
$750.00 sale price
15 lb material
20 hr labor
20 unit demand per week
supply per week
500 lb material
400 hr labor = 10 employee x 40 hr/employee
extra
$9.00/lb material [p1]
$22.50/hr labor [p2] ($22.50/hr overtime = 1.5 x $15.00/hr normal) (capped at 40 additional hours)
initial cost D
$40,800.00 = $28 .00/hr x 1000 hr + $4 .00/lb x 3200 lb
[A] What number of Type A and Type B valves maximizes weekly profit, without scheduling overtime or purchasing additional material?
[B] Is it profitable for the company to purchase additional material, without scheduling overtime? If so, how much additional material?
[C] Is it profitable for the company to schedule overtime, given the result of [B]? If so, how much overtime?
[D] Assuming the company has access to a given week's revenue minus costs incurred for additional resources prior to the start of the next week, how much additional capital does the company need to maximize profit?
[E] If the company only has $5,000.00 in capital for the purposes of increasing production (and not the amount determined to meet demand in [D]), what allocation of overtime and additional material maximizes the company's profit?
[F] Assuming the result of [C], what is the change in optimal revenue per unit
increase in material
decrease in material
increase in labor
decrease in labor
increase in demand for Type A
decrease in demand for Type A
increase in demand for Type B
decrease in demand for Type B
[G] What are the rates of change in [F] at the primal solution to [C]; to [D]?
[H] What is the change in the optimal profit per dollar capital at the solution to [E]?
\(\text{LP \& LP}^*\)#
\( \boxed{ \text{LP}= \begin{aligned} \max z(\mathbf{x})&=\mathbf{cx}+d \\ \mathbf{Ax}&\le\mathbf{b} \\ \mathbf{x}&\ge\mathbf{0} \end{aligned} \overset{\text{dual to}}{\iff} \begin{aligned} \min w(\mathbf{y})&=\mathbf{yb}+d \\ \mathbf{yA}&\ge\mathbf{c} \\ \mathbf{y}&\ge\mathbf{0} \end{aligned} =\text{LP}^* } \)
\( \text{LP}= \begin{aligned} \max z(\mathbf{x})&=500x_1+750x_2-d \\ g_1(\mathbf{x})&=20x_1+15x_2\le500=b_1 &&\text{material constraint} \\ g_2(\mathbf{x})&=10x_1+20x_2\le400=b_2 &&\text{labor constraint} \\ g_3(\mathbf{x})&=x_1\le25=b_3 &&\text{Type A valve demand constaint} \\ g_4(\mathbf{x})&=x_2\le20=b_4 &&\text{Type B valve demand constraint} \\ \mathbf{x}&\ge\mathbf{0} &&\text{no negative production} \end{aligned} \overset{\text{dual to}}{\iff} \begin{aligned} \min w(\mathbf{y})&=500y_1+400y_2+25y_3+20y_4-d \\ h_1(\mathbf{y})&= 20y_1+ 10y_2+ y_3 \ge500=c_1\\ h_2(\mathbf{y})&= 15y_1+ 20y_2+ y_4 \ge750=c_2\\ \mathbf{y}&\ge\mathbf{0} \end{aligned} =\text{LP}^* \)
LP A B
max z = 500x1 + 750x2 - d
g1 = 20x1 + 15x2 <= 500 = b1 material constraint
g2 = 10x1 + 20x2 <= 400 = b2 labor constraint
g3 = 1x1 + 0x2 <= 25 = b3 Type A valve demand constraint
g4 = 0x1 + 1x2 <= 20 = b4 Type B valve demand constraint
x >= 0 no negative production
d = 0
b1=500
b2=400
b3= 25
b4= 20
z =500*x1+750*x2- d
g1= 20*x1+ 15*x2-b1
g2= 10*x1+ 20*x2-b2
g3= 1*x1+ 0*x2-b3
g4= 0*x1+ 1*x2-b4
db1=0
db2=0
db3=0
db4=0
p1= 9.00
p2=22.50
A=np.array([[20,15],[10,20],[1,0],[0,1]])
b=np.array([500,400,25,20])
c=np.array([500,750])
d=np.array([0])
x=np.array([x1,x2])
y=np.array([y1,y2,y3,y4])
display(Math(
r"\begin{aligned}"
fr"\max z&={latex( c@x+d [0])}\\"
fr" g_1&={latex((A@x-b)[0])}\\"
fr" g_2&={latex((A@x-b)[1])}\\"
fr" g_3&={latex((A@x-b)[2])}\\"
fr" g_4&={latex((A@x-b)[3])}\\"
r"\\"
fr"\min w&={latex( y@b+d [0])}\\"
fr" h_1&={latex((y@A-c)[0])}\\"
fr" h_2&={latex((y@A-c)[1])}\\"
r"\end{aligned}"),
)
\[\begin{split}\displaystyle \begin{aligned}\max z&=500 x_{1} + 750 x_{2}\\ g_1&=20 x_{1} + 15 x_{2} - 500\\ g_2&=10 x_{1} + 20 x_{2} - 400\\ g_3&=x_{1} - 25\\ g_4&=x_{2} - 20\\\\\min w&=500 y_{1} + 400 y_{2} + 25 y_{3} + 20 y_{4}\\ h_1&=20 y_{1} + 10 y_{2} + y_{3} - 500\\ h_2&=15 y_{1} + 20 y_{2} + y_{4} - 750\\\end{aligned}\end{split}\]
Gradients and directions#
\( \begin{aligned} \nabla z&=\langle500,750\rangle \\ \nabla g_1&=\langle20,15\rangle &&\langle500,750\rangle\cdot\langle15,-20\rangle &&\lt0 &&\implies\langle-15,20\rangle \\ \nabla g_2&=\langle10,20\rangle &&\langle500,750\rangle\cdot\langle20,-10\rangle &&\gt0 &&\implies\langle20,-10\rangle \\ \nabla g_3&=\langle1,0\rangle &&\langle500,750\rangle\cdot\langle0,-1\rangle &&\lt0 &&\implies\langle0,1\rangle \\ \nabla g_4&=\langle0,1\rangle &&\langle500,750\rangle\cdot\langle1,0\rangle &&\gt0 &&\implies\langle1,0\rangle \end{aligned} \)
delz = <500,750>
delg1 = < 20, 15>
delg2 = < 10, 20>
delg3 = < 1, 0>
delg4 = < 0, 1>
<500,750><15,-20> <0 => <-15, 20> g1
<500,750><20,-10> >0 => < 20,-10> g2
<500,750>< 0, -1> <0 => < 0, 1> g3
<500,750>< 1, 0> >0 => < 1, 0> g4
delz =c
delg1=A[0] ; dg1=(delg1[1],-delg1[0]) if delz@(delg1[1],-delg1[0]) > 0 else (-delg1[1],delg1[0])
delg2=A[1] ; dg2=(delg2[1],-delg2[0]) if delz@(delg2[1],-delg2[0]) > 0 else (-delg2[1],delg2[0])
delg3=A[2] ; dg3=(delg3[1],-delg3[0]) if delz@(delg3[1],-delg3[0]) > 0 else (-delg3[1],delg3[0])
delg4=A[3] ; dg4=(delg4[1],-delg4[0]) if delz@(delg4[1],-delg4[0]) > 0 else (-delg4[1],delg4[0])
display(Math(
r"\begin{aligned}"
fr"\nabla z &=\langle{ delz[0]},{ delz[1]}\rangle\\"
fr"\nabla g_1 &=\langle{delg1[0]},{delg1[1]}\rangle && dg_1=\langle{dg1[0]},{dg1[1]}\rangle\\"
fr"\nabla g_2 &=\langle{delg2[0]},{delg2[1]}\rangle && dg_2=\langle{dg2[0]},{dg2[1]}\rangle\\"
fr"\nabla g_3 &=\langle{delg3[0]},{delg3[1]}\rangle && dg_3=\langle{dg3[0]},{dg3[1]}\rangle\\"
fr"\nabla g_4 &=\langle{delg4[0]},{delg4[1]}\rangle && dg_4=\langle{dg4[0]},{dg4[1]}\rangle\\"
r"\end{aligned}"),
)
\[\begin{split}\displaystyle \begin{aligned}\nabla z &=\langle500,750\rangle\\\nabla g_1 &=\langle20,15\rangle && dg_1=\langle-15,20\rangle\\\nabla g_2 &=\langle10,20\rangle && dg_2=\langle20,-10\rangle\\\nabla g_3 &=\langle1,0\rangle && dg_3=\langle0,1\rangle\\\nabla g_4 &=\langle0,1\rangle && dg_4=\langle1,0\rangle\\\end{aligned}\end{split}\]
# UNIT DIRECTION VECTORS
udg1=dg1/np.sqrt(dg1[0]**2+dg1[1]**2)
udg2=dg2/np.sqrt(dg2[0]**2+dg2[1]**2)
udg3=dg3/np.sqrt(dg3[0]**2+dg3[1]**2)
udg4=dg4/np.sqrt(dg4[0]**2+dg4[1]**2)
display(Math(
r"\begin{aligned}"
r"\hat{dg_1}&="
fr"\langle{udg1[0]},{udg1[1]}\rangle\\"
r"\hat{dg_2}&="
fr"\langle{udg2[0]},{udg2[1]}\rangle\\"
r"\hat{dg_3}&="
fr"\langle{udg3[0]},{udg3[1]}\rangle\\"
r"\hat{dg_4}&="
fr"\langle{udg4[0]},{udg4[1]}\rangle\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}\hat{dg_1}&=\langle-0.6,0.8\rangle\\\hat{dg_2}&=\langle0.8944271909999159,-0.4472135954999579\rangle\\\hat{dg_3}&=\langle0.0,1.0\rangle\\\hat{dg_4}&=\langle1.0,0.0\rangle\\\end{aligned}\end{split}\]
Simplex Algorithm#
# CONCISE ROW TABLEAU
rA = Matrix([
[ 'c', 'x1', 'x2', 1, 'r'],
['y1', -20, -15,b1+db1,'s1'],
['y2', -10, -20,b2+db2,'s2'],
['y3', -1, 0,b3+db3,'s3'],
['y4', 0, -1,b4+db4,'s4'],
[ -1, -500, -750, -d[0],'-z'],
[ 'c','-v1','-v2', 'w', 'r'],
])
rB = pivot(1,1,rA)
rC = pivot(2,2,rB)
# CONCISE COLUMN TABLEAU
cA = Matrix([
[ 'c', 'y1', 'y2', 'y3', 'y4', 1, 'r'],
['x1', 20, 10, 1, 0,-500,'v1'],
['x2', 15, 20, 0, 1,-750,'v2'],
[ -1,b1+db1,b2+db2,b3+db3,b4+db4,d[0], 'w'],
[ 'c', '-s1', '-s2', '-s3', '-s4','-z', 'r'],
])
cB = pivot(1,1,cA)
cC = pivot(2,2,cB)
display(Math(
r'\begin{aligned}'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rA = {latex(rA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rB = {latex(rB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rC = {latex(rC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cA = {latex(cA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cB = {latex(cB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cC = {latex(cC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
r'\end{aligned}'
))
\[\begin{split}\displaystyle \begin{aligned}db &= (0,0,0,0)\,\,\,rA = \left[\begin{matrix}c & x_{1} & x_{2} & 1 & r\\y_{1} & -20 & -15 & 500 & s_{1}\\y_{2} & -10 & -20 & 400 & s_{2}\\y_{3} & -1 & 0 & 25 & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & -500 & -750 & 0 & - z\\c & - v_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (0,0,0,0)\,\,\,rB = \left[\begin{matrix}c & s_{1} & x_{2} & 1 & r\\v_{1} & - \frac{1}{20} & - \frac{3}{4} & 25 & x_{1}\\y_{2} & \frac{1}{2} & - \frac{25}{2} & 150 & s_{2}\\y_{3} & \frac{1}{20} & \frac{3}{4} & 0 & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & 25 & -375 & -12500 & - z\\c & - y_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (0,0,0,0)\,\,\,rC = \left[\begin{matrix}c & s_{1} & s_{2} & 1 & r\\v_{1} & - \frac{2}{25} & \frac{3}{50} & 16 & x_{1}\\v_{2} & \frac{1}{25} & - \frac{2}{25} & 12 & x_{2}\\y_{3} & \frac{2}{25} & - \frac{3}{50} & 9 & s_{3}\\y_{4} & - \frac{1}{25} & \frac{2}{25} & 8 & s_{4}\\-1 & 10 & 30 & -17000 & - z\\c & - y_{1} & - y_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (0,0,0,0)\,\,\,cA = \left[\begin{matrix}c & y_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\x_{1} & 20 & 10 & 1 & 0 & -500 & v_{1}\\x_{2} & 15 & 20 & 0 & 1 & -750 & v_{2}\\-1 & 500 & 400 & 25 & 20 & 0 & w\\c & - s_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (0,0,0,0)\,\,\,cB = \left[\begin{matrix}c & v_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{1}{20} & - \frac{1}{2} & - \frac{1}{20} & 0 & 25 & y_{1}\\x_{2} & \frac{3}{4} & \frac{25}{2} & - \frac{3}{4} & 1 & -375 & v_{2}\\-1 & 25 & 150 & 0 & 20 & 12500 & w\\c & - x_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (0,0,0,0)\,\,\,cC = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{2}{25} & - \frac{1}{25} & - \frac{2}{25} & \frac{1}{25} & 10 & y_{1}\\s_{2} & - \frac{3}{50} & \frac{2}{25} & \frac{3}{50} & - \frac{2}{25} & 30 & y_{2}\\-1 & 16 & 12 & 9 & 8 & 17000 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\\end{aligned}\end{split}\]
Show code cell source
xStar0=(16,12)
zStar0=z.subs([(x1,xStar0[0]),(x2,xStar0[1])])
yStar =(10,30,0,0)
ROI1 =round((yStar[0]-p1)/p1,2)
ROI2 =round((yStar[1]-p2)/p2,2)
display(Math(
r"\begin{aligned}"
r"x^*&="
f"{xStar0}"
r"\\"
r"z^*&="
f"\${int(zStar0):,.2f}"
f"&&={latex(z)}"
r"=w^*"
r"\\"
r"y^*&="
f"{yStar}"
r"\\"
r"\text{ROI}_\text{material}&="
f"{ROI1}"
r"&&=\frac{y_1^*-p_1}{p_1}"
r"\\"
r"\text{ROI}_\text{labor}&="
f"{ROI2}"
r"&&=\frac{y_2^*-p_2}{p_2}"
r"\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}x^*&=(16, 12)\\z^*&=\$17,000.00&&=500 x_{1} + 750 x_{2}=w^*\\y^*&=(10, 30, 0, 0)\\\text{ROI}_\text{material}&=0.11&&=\frac{y_1^*-p_1}{p_1}\\\text{ROI}_\text{labor}&=0.33&&=\frac{y_2^*-p_2}{p_2}\\\end{aligned}\end{split}\]
Graphical#
display(
'LINES',
Math(f"g_1 : {latex(solve(g1,x2,dict=True)[0])}"),
Math(f"g_2 : {latex(solve(g2,x2,dict=True)[0])}"),
Math(f"g_4 : {latex(solve(g4,x2,dict=True)[0])}"),
'FEASIBLE SIDES',
Math(f"g_1 : {latex(reduce_inequalities(g1<=0,[x2]))}"),
Math(f"g_2 : {latex(reduce_inequalities(g2<=0,[x2]))}"),
Math(f"g_3 : {latex(reduce_inequalities(g3<=0,[x2]))}"),
Math(f"g_4 : {latex(reduce_inequalities(g4<=0,[x2]))}"),
'INTERSECTIONS',
Math(f"g_1=g_2 : {latex(solve([g1,g2],[x1,x2],dict=True)[0])}"),
Math(f"g_1=g_3 : {latex(solve([g1,g3],[x1,x2],dict=True)[0])}"),
Math(f"g_1=g_4 : {latex(solve([g1,g4],[x1,x2],dict=True)[0])}"),
Math(f"g_2=g_3 : {latex(solve([g2,g3],[x1,x2],dict=True)[0])}"),
Math(f"g_2=g_4 : {latex(solve([g2,g4],[x1,x2],dict=True)[0])}"),
Math(f"g_3=g_4 : {latex(solve([g3,g4],[x1,x2],dict=True)[0])}"),
Math(f"g_1[ 0,x2] : {latex(solve([g1,x1],[x1,x2],dict=True)[0])}"),
Math(f"g_1[x1, 0] : {latex(solve([g1,x2],[x1,x2],dict=True)[0])}"),
Math(f"g_2[ 0,x2] : {latex(solve([g2,x1],[x1,x2],dict=True)[0])}"),
Math(f"g_2[x1, 0] : {latex(solve([g2,x2],[x1,x2],dict=True)[0])}"),
Math(f"g_3[x1, 0] : {latex(solve([g3,x2],[x1,x2],dict=True)[0])}"),
Math(f"g_4[ 0,x2] : {latex(solve([g4,x1],[x1,x2],dict=True)[0])}"),
)
'LINES'
\[\displaystyle g_1 : \left\{ x_{2} : \frac{100}{3} - \frac{4 x_{1}}{3}\right\}\]
\[\displaystyle g_2 : \left\{ x_{2} : 20 - \frac{x_{1}}{2}\right\}\]
\[\displaystyle g_4 : \left\{ x_{2} : 20\right\}\]
'FEASIBLE SIDES'
\[\displaystyle g_1 : x_{2} \leq \frac{100}{3} - \frac{4 x_{1}}{3} \wedge -\infty < x_{2}\]
\[\displaystyle g_2 : x_{2} \leq 20 - \frac{x_{1}}{2} \wedge -\infty < x_{2}\]
\[\displaystyle g_3 : x_{1} \leq 25 \wedge -\infty < x_{1}\]
\[\displaystyle g_4 : x_{2} \leq 20 \wedge -\infty < x_{2}\]
'INTERSECTIONS'
\[\displaystyle g_1=g_2 : \left\{ x_{1} : 16, \ x_{2} : 12\right\}\]
\[\displaystyle g_1=g_3 : \left\{ x_{1} : 25, \ x_{2} : 0\right\}\]
\[\displaystyle g_1=g_4 : \left\{ x_{1} : 10, \ x_{2} : 20\right\}\]
\[\displaystyle g_2=g_3 : \left\{ x_{1} : 25, \ x_{2} : \frac{15}{2}\right\}\]
\[\displaystyle g_2=g_4 : \left\{ x_{1} : 0, \ x_{2} : 20\right\}\]
\[\displaystyle g_3=g_4 : \left\{ x_{1} : 25, \ x_{2} : 20\right\}\]
\[\displaystyle g_1[ 0,x2] : \left\{ x_{1} : 0, \ x_{2} : \frac{100}{3}\right\}\]
\[\displaystyle g_1[x1, 0] : \left\{ x_{1} : 25, \ x_{2} : 0\right\}\]
\[\displaystyle g_2[ 0,x2] : \left\{ x_{1} : 0, \ x_{2} : 20\right\}\]
\[\displaystyle g_2[x1, 0] : \left\{ x_{1} : 40, \ x_{2} : 0\right\}\]
\[\displaystyle g_3[x1, 0] : \left\{ x_{1} : 25, \ x_{2} : 0\right\}\]
\[\displaystyle g_4[ 0,x2] : \left\{ x_{1} : 0, \ x_{2} : 20\right\}\]
s =40
x =np.linspace(0,s,1001)
# slopes
s1=-(4/3)
s2=-(1/2)
s4=0
y1=s1*x+(100/3)
y2=s2*x+( 20/1)
y4=s4*x+20
Show code cell source
fig=plt.figure(dpi=300);
##########
ax =plt.subplot();
ax.set_aspect(1);
# LINES
ax.plot (x,y1,label='g1=b1',linewidth=0.5,color='blue');
ax.plot (x,y2,label='g2=b2',linewidth=0.5,color='yellow');
ax.axvline( 25,label='g3=b3',linewidth=0.5,color='green');
ax.plot (x,y4,label='g4=b4',linewidth=0.5,color='red');
# DELZ
ax.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*fx+(100/3))
& (fy<=s2*fx+( 20/1))
& (fy<=s4*fx+20 )
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax.set_xlim(0,s);
ax.set_ylim(0,s);
ax.legend();
# XSTAR
ax.scatter(*xStar0,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&{xStar0}'
r'\\'
f'z^*&=&\${round(float(zStar0),2):,.2f}=w^*'
r'\\'
f'y^*&=&{yStar}'
r'\\'
r'\mathrm{ROI}_\mathrm{material}'
f'&=&{ROI1}'
r'\\'
r'\mathrm{ROI}_\mathrm{labor}'
f'&=&{ROI2}'
r'\end{eqnarray*}'
)
ax.text(10,35,sol,fontsize=10);
# SIMPLEX
dx,dy=1,1
ax.annotate('$A$',( 0+dx, 0+dy));
ax.annotate('$B$',( 25+dx, 0+dy));
ax.annotate('$C$',(xStar0[0]+dx,xStar0[1]+dy));
Error in callback <function _draw_all_if_interactive at 0x1302e8e00> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/formatters.py:343, in BaseFormatter.__call__(self, obj)
341 pass
342 else:
--> 343 return printer(obj)
344 # Finally look for special method names
345 method = get_real_method(obj, self.print_method)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
<Figure size 1920x1440 with 1 Axes>
[B] Is it profitable for the company to purchase additional material?#
\( \begin{aligned} \text{change in revenue per unit change in material}\,\,\, \partial z^*&=y_1^*\partial b_1 \impliedby y_1^*=\frac{\partial z^*}{\partial b_1} \\ \text{profit}\,\,\, &=\text{change in revenue}-\text{cost of material} \\ &=y_1^*\partial b_1-p_1\partial b_1 \\ &=(y_1^*-p_1)\partial b_1 \end{aligned} \)
Show code cell source
g2g3 =solve([g2,g3])
xStar1=(g2g3[x1],g2g3[x2])
g1n =s1*(x1-xStar1[0])+xStar1[1]
b1n =Fraction(float(g1.subs([(x1,xStar1[0]),(x2,xStar1[1])])+b1)).limit_denominator(2)
db1 =b1n-b1
rc =p1*db1
zStar1=z.subs([(x1,xStar1[0]),(x2,xStar1[1])])-rc
dzStar=yStar[0]*db1-rc
# db1=symbols('db1')
# solve([
# g1.subs([(x1,25)])-db1,
# g2.subs([(x1,25)]),
# ],[db1,x2])
display(Math(
r"\begin{aligned}"
r"y_1^*\overset{?}{\gt}p_1&\implies"
f"{yStar[0]}"
r"\overset{?}{\gt}"
f"{p1}"
r"\\"
r"g_2=g_3&\implies"
f"{latex(g2g3)}"
r"\\"
r"x^*(\text{new})&="
f"{xStar1}"
r"\\"
r"g_1(\text{new})&="
f"{latex(nsimplify(g1n))}"
r"\\"
r"g_1(\text{new})=x_2&\implies x_1="
f"{latex(nsimplify(solve([g1n,x2])[x1]))}"
r"\\"
r" b_1(\text{new})&="
f"{b1n}"
r"\\"
fr"db_1&={db1}"
r"&&=b_1(\text{new})-b_1"
r"\\"
r"\text{cost}_\text{material}&="
f"\${rc:,.2f}"
r"&&=p_1\partial b_1"
r"\\"
r"z^*(\text{new})&="
f"\${zStar1:,.2f}"
f"&&={latex(z-rc)}"
r"=w^*\\"
r"\partial z^*&="
f"\${dzStar:,.2f}"
r"&&=y_1^*\partial b_1=z^*(\text{new})-z^*"
r"\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}y_1^*\overset{?}{\gt}p_1&\implies10\overset{?}{\gt}9.0\\g_2=g_3&\implies\left\{ x_{1} : 25, \ x_{2} : \frac{15}{2}\right\}\\x^*(\text{new})&=(25, 15/2)\\g_1(\text{new})&=\frac{245}{6} - \frac{4 x_{1}}{3}\\g_1(\text{new})=x_2&\implies x_1=\frac{245}{8}\\ b_1(\text{new})&=1225/2\\db_1&=225/2&&=b_1(\text{new})-b_1\\\text{cost}_\text{material}&=\$1,012.50&&=p_1\partial b_1\\z^*(\text{new})&=\$17,112.50&&=500 x_{1} + 750 x_{2} - 1012.5=w^*\\\partial z^*&=\$112.50&&=y_1^*\partial b_1=z^*(\text{new})-z^*\\\end{aligned}\end{split}\]
Show code cell source
fig=plt.figure(dpi=300);
##########
ax =plt.subplot();
ax.set_aspect(1);
# LINES
ax.plot (x,y1, label='g1=b1', linewidth=0.5,linestyle=(0,(5,10)),color='blue');
ax.plot (x,s1*(x-xStar1[0])+xStar1[1],label='g1=b1+db1',linewidth=0.5, color='blue');
ax.plot (x,y2, label='g2=b2', linewidth=0.5, color='yellow');
ax.axvline( 25, label='g3=b3', linewidth=0.5, color='green');
ax.plot (x,y4, label='g4=b4', linewidth=0.5, color='red');
# DELZ
ax.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*(fx-float(xStar1[0]))+float(xStar1[1]))
& (fy<=s2*fx+20)
& (fx<=25)
& (fy<= 0*fx+20)
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax.set_xlim(0,s);
ax.set_ylim(0,s);
ax.legend();
# XSTAR
ax.scatter(*xStar1,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&{xStar1}'
r'\\'
f'z^*&=&\${round(zStar1,2):,.2f}=w^*'
r'\\'
r'\end{eqnarray*}'
)
ax.text(10,35,sol,fontsize=10);
# SIMPLEX
# dx,dy=1,1
# ax.annotate('$A$',( 0+dx, 0+dy));
# ax.annotate('$B$',( (245/8)+dx, 0+dy));
# ax.annotate('$C$',(xStar1[0]+dx,xStar1[1]+dy));
Error in callback <function _draw_all_if_interactive at 0x1302e8e00> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/formatters.py:343, in BaseFormatter.__call__(self, obj)
341 pass
342 else:
--> 343 return printer(obj)
344 # Finally look for special method names
345 method = get_real_method(obj, self.print_method)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
<Figure size 1920x1440 with 1 Axes>
[C] Is it profitable for the company to schedule overtime, having purchased additional material?#
Show code cell source
d =d-rc
# CONCISE ROW TABLEAU
rA = Matrix([
[ 'c', 'x1', 'x2', 1, 'r'],
['y1', -20, -15,b1+db1,'s1'],
['y2', -10, -20,b2+db2,'s2'],
['y3', -1, 0,b3+db3,'s3'],
['y4', 0, -1,b4+db4,'s4'],
[ -1, -500, -750, -d[0],'-z'],
[ 'c','-v1','-v2', 'w', 'r'],
])
rB = pivot(1,1,rA)
rC = pivot(2,2,rB)
rD = pivot(3,1,rC)
# CONCISE COLUMN TABLEAU
cA = Matrix([
[ 'c', 'y1', 'y2', 'y3', 'y4', 1, 'r'],
['x1', 20, 10, 1, 0,-500,'v1'],
['x2', 15, 20, 0, 1,-750,'v2'],
[ -1,b1+db1,b2+db2,b3+db3,b4+db4,d[0], 'w'],
[ 'c', '-s1', '-s2', '-s3', '-s4','-z', 'r'],
])
cB = pivot(1,1,cA)
cC = pivot(2,2,cB)
cD = pivot(1,3,cC)
display(Math(
r'\begin{aligned}'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rA = {latex(rA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rB = {latex(rB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rC = {latex(rC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rD = {latex(rD)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_2=s_3=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cA = {latex(cA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cB = {latex(cB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cC = {latex(cC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cD = {latex(cD)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_2=s_3=0\end{aligned}\\'
r'\end{aligned}'
))
\[\begin{split}\displaystyle \begin{aligned}db &= (225/2,0,0,0)\,\,\,rA = \left[\begin{matrix}c & x_{1} & x_{2} & 1 & r\\y_{1} & -20 & -15 & \frac{1225}{2} & s_{1}\\y_{2} & -10 & -20 & 400 & s_{2}\\y_{3} & -1 & 0 & 25 & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & -500 & -750 & 1012.5 & - z\\c & - v_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (225/2,0,0,0)\,\,\,rB = \left[\begin{matrix}c & s_{1} & x_{2} & 1 & r\\v_{1} & - \frac{1}{20} & - \frac{3}{4} & \frac{245}{8} & x_{1}\\y_{2} & \frac{1}{2} & - \frac{25}{2} & \frac{375}{4} & s_{2}\\y_{3} & \frac{1}{20} & \frac{3}{4} & - \frac{45}{8} & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & 25 & -375 & -14300.0 & - z\\c & - y_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (225/2,0,0,0)\,\,\,rC = \left[\begin{matrix}c & s_{1} & s_{2} & 1 & r\\v_{1} & - \frac{2}{25} & \frac{3}{50} & 25 & x_{1}\\v_{2} & \frac{1}{25} & - \frac{2}{25} & \frac{15}{2} & x_{2}\\y_{3} & \frac{2}{25} & - \frac{3}{50} & 0 & s_{3}\\y_{4} & - \frac{1}{25} & \frac{2}{25} & \frac{25}{2} & s_{4}\\-1 & 10 & 30 & -17112.5 & - z\\c & - y_{1} & - y_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (225/2,0,0,0)\,\,\,rD = \left[\begin{matrix}c & s_{3} & s_{2} & 1 & r\\v_{1} & -1 & 0 & 25 & x_{1}\\v_{2} & \frac{1}{2} & - \frac{1}{20} & \frac{15}{2} & x_{2}\\y_{1} & \frac{25}{2} & \frac{3}{4} & 0 & s_{1}\\y_{4} & - \frac{1}{2} & \frac{1}{20} & \frac{25}{2} & s_{4}\\-1 & 125 & \frac{75}{2} & -17112.5 & - z\\c & - y_{3} & - y_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_2=s_3=0\end{aligned}\\db &= (225/2,0,0,0)\,\,\,cA = \left[\begin{matrix}c & y_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\x_{1} & 20 & 10 & 1 & 0 & -500 & v_{1}\\x_{2} & 15 & 20 & 0 & 1 & -750 & v_{2}\\-1 & \frac{1225}{2} & 400 & 25 & 20 & -1012.5 & w\\c & - s_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (225/2,0,0,0)\,\,\,cB = \left[\begin{matrix}c & v_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{1}{20} & - \frac{1}{2} & - \frac{1}{20} & 0 & 25 & y_{1}\\x_{2} & \frac{3}{4} & \frac{25}{2} & - \frac{3}{4} & 1 & -375 & v_{2}\\-1 & \frac{245}{8} & \frac{375}{4} & - \frac{45}{8} & 20 & 14300.0 & w\\c & - x_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (225/2,0,0,0)\,\,\,cC = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{2}{25} & - \frac{1}{25} & - \frac{2}{25} & \frac{1}{25} & 10 & y_{1}\\s_{2} & - \frac{3}{50} & \frac{2}{25} & \frac{3}{50} & - \frac{2}{25} & 30 & y_{2}\\-1 & 25 & \frac{15}{2} & 0 & \frac{25}{2} & 17112.5 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (225/2,0,0,0)\,\,\,cD = \left[\begin{matrix}c & v_{1} & v_{2} & y_{1} & y_{4} & 1 & r\\s_{3} & 1 & - \frac{1}{2} & - \frac{25}{2} & \frac{1}{2} & 125 & y_{3}\\s_{2} & 0 & \frac{1}{20} & - \frac{3}{4} & - \frac{1}{20} & \frac{75}{2} & y_{2}\\-1 & 25 & \frac{15}{2} & 0 & \frac{25}{2} & 17112.5 & w\\c & - x_{1} & - x_{2} & - s_{1} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_2=s_3=0\end{aligned}\\\end{aligned}\end{split}\]
Show code cell source
ycStar1=yStar
ydStar1=(0,75/2,125,0)
ROI1c=round((ycStar1[0]-p1)/p1,2)
ROI2c=round((ycStar1[1]-p2)/p2,2)
ROI1d=round((ydStar1[0]-p1)/p1,2)
ROI2d=round((ydStar1[1]-p2)/p2,2)
display(Math(
r"\begin{aligned}"
r"x^*&="
f"{xStar1}"
r"\\"
r"z^*&="
f"\${zStar1:,.2f}"
f"&&={latex(z-rc)}"
r"=w^*\\"
r"yc^*&="
f"{ycStar1}"
r"&&=\left(yc_1^*=\frac{\partial z^*}{\partial b_1},yc_2^*=\frac{\partial z^*}{\partial b_2},yc_3^*=\frac{\partial z^*}{\partial b_3},yc_4^*=\frac{\partial z^*}{\partial b_4}\right)"
r"\\"
r"\text{ROI}_\text{material}&="
f"{ROI1c}"
r"&&=\frac{yc_1^*-p_1}{p_1}"
r"\\"
r"\text{ROI}_\text{labor}&="
f"{ROI2c}"
r"&&=\frac{yc_2^*-p_2}{p_2}"
r"\\"
r"yd^*&="
f"{ydStar1}"
r"&&=\left(yd_1^*=\frac{\partial z^*}{\partial b_1},yd_2^*=\frac{\partial z^*}{\partial b_2},yd_3^*=\frac{\partial z^*}{\partial b_3},yd_4^*=\frac{\partial z^*}{\partial b_4}\right)"
r"\\"
r"\text{ROI}_\text{material}&="
f"{ROI1d}"
r"&&=\frac{yd_1^*-p_1}{p_1}"
r"\\"
r"\text{ROI}_\text{labor}&="
f"{ROI2d}"
r"&&=\frac{yd_2^*-p_2}{p_2}"
r"\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}x^*&=(25, 15/2)\\z^*&=\$17,112.50&&=500 x_{1} + 750 x_{2} - 1012.5=w^*\\yc^*&=(10, 30, 0, 0)&&=\left(yc_1^*=\frac{\partial z^*}{\partial b_1},yc_2^*=\frac{\partial z^*}{\partial b_2},yc_3^*=\frac{\partial z^*}{\partial b_3},yc_4^*=\frac{\partial z^*}{\partial b_4}\right)\\\text{ROI}_\text{material}&=0.11&&=\frac{yc_1^*-p_1}{p_1}\\\text{ROI}_\text{labor}&=0.33&&=\frac{yc_2^*-p_2}{p_2}\\yd^*&=(0, 37.5, 125, 0)&&=\left(yd_1^*=\frac{\partial z^*}{\partial b_1},yd_2^*=\frac{\partial z^*}{\partial b_2},yd_3^*=\frac{\partial z^*}{\partial b_3},yd_4^*=\frac{\partial z^*}{\partial b_4}\right)\\\text{ROI}_\text{material}&=-1.0&&=\frac{yd_1^*-p_1}{p_1}\\\text{ROI}_\text{labor}&=0.67&&=\frac{yd_2^*-p_2}{p_2}\\\end{aligned}\end{split}\]
Show code cell source
g1ng4 =solve([g1n-x2,g4])
#g1ng4=solve([g1-db1,g4])
xStar2=(g1ng4[x1],g1ng4[x2])
g2n =s2*(x1-xStar2[0])+xStar2[1]
b2n =float(g2.subs([(x1,xStar2[0]),(x2,xStar2[1])])+b2)
db2 =b2n-b2
rc2 =p2*db2
zStar2=z.subs([(x1,xStar2[0]),(x2,xStar2[1])])-rc2-rc
dzStar=yStar[0]*db2
# db1=symbols('db1')
# solve([
# g1.subs([(x1,25)])-db1,
# g2.subs([(x1,25)]),
# ],[db1,x2])
display(Math(
r"\begin{aligned}"
r"y_2^*\overset{?}{\gt}p_2&\implies"
f"{yStar[1]}"
r"\overset{?}{\gt}"
f"{p2}"
r"\\"
r"g_1(\text{new})=g_4&\implies"
f"{latex(g1ng4)}"
r"\\"
r"x^*(\text{new})&="
f"{nsimplify(xStar2[0]),nsimplify(xStar2[1])}"
r"\\"
r"g_2(\text{new})&="
f"{latex(nsimplify(g2n))}"
r"\\"
r"g_2(\text{new})=x_1&\implies x_1="
# f"{latex(nsimplify(solve([g2n,x1])[x2]))}"
r"\\"
r" b_2(\text{new})&="
f"{b2n}"
r"\\"
fr"db_2&={db2}"
r"&&=b_2(\text{new})-b_2"
r"\\"
r"\text{cost}&="
f"\${rc2:,.2f}"
r"&&=p_2\partial b_2"
r"\\"
r"z^*(\text{new})&="
f"\${zStar2:,.2f}"
f"&&={latex(z-rc2)}"
r"=w^*\\"
r"\partial z^*&="
f"\${dzStar:,.2f}"
r"&&=y_2^*\partial b_2=z^*(\text{new})-z^*"
r"\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}y_2^*\overset{?}{\gt}p_2&\implies30\overset{?}{\gt}22.5\\g_1(\text{new})=g_4&\implies\left\{ x_{1} : 15.625, \ x_{2} : 20.0\right\}\\x^*(\text{new})&=(125/8, 20)\\g_2(\text{new})&=\frac{445}{16} - \frac{x_{1}}{2}\\g_2(\text{new})=x_1&\implies x_1=\\ b_2(\text{new})&=556.25\\db_2&=156.25&&=b_2(\text{new})-b_2\\\text{cost}&=\$3,515.62&&=p_2\partial b_2\\z^*(\text{new})&=\$18,284.38&&=500 x_{1} + 750 x_{2} - 3515.625=w^*\\\partial z^*&=\$1,562.50&&=y_2^*\partial b_2=z^*(\text{new})-z^*\\\end{aligned}\end{split}\]
Show code cell source
fig=plt.figure(dpi=300,figsize=(8,16));
##########
ax =plt.subplot(121);
ax.set_aspect(1);
# LINES
ax.plot (x,y1, label='g1=b1', linewidth=0.5,linestyle=(0,(5,10)),color='blue');
ax.plot (x,s1*(x-xStar1[0])+xStar1[1],label='g1=b1+db1',linewidth=0.5, color='blue');
ax.plot (x,y2, label='g2=b2', linewidth=0.5, color='yellow');
ax.axvline( 25, label='g3=b3', linewidth=0.5, color='green');
ax.plot (x,y4, label='g4=b4', linewidth=0.5, color='red');
# DELZ
ax.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*(fx-float(xStar1[0]))+float(xStar1[1]))
& (fy<=s2*fx+20)
& (fx<=25)
& (fy<= 0*fx+20)
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax.set_xlim(0,s);
ax.set_ylim(0,s);
#ax.legend();
# XSTAR
ax.scatter(*xStar1,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&{xStar1}'
r'\\'
f'z^*&=&\${round(zStar1,2):,.2f}=w^*'
r'\\'
f'yc^*&=&{ycStar1}'
r'\\'
f'yd^*&=&{ydStar1}'
r'\\'
r'\mathrm{ROI}_\mathrm{material}'
f'&=&{ROI1d}'
r'\\'
r'\mathrm{ROI}_\mathrm{labor}'
f'&=&{ROI2d}'
r'\end{eqnarray*}'
)
ax.text(10,35,sol,fontsize=10);
# SIMPLEX
dx,dy=1,1
ax.annotate('$A$',( 0+dx, 0+dy));
ax.annotate('$B$',( (245/8)+dx, 0+dy));
ax.annotate('$C$',(xStar1[0]+dx,xStar1[1]+dy));
##########
ax2 =plt.subplot(122);
ax2.set_aspect(1);
# LINES
ax2.plot (x,y1, label='g1=b1', linewidth=0.5,linestyle=(0,(5,10)),color='blue');
ax2.plot (x,s1*(x-xStar1[0])+xStar1[1],label='g1=b1+db1',linewidth=0.5, color='blue');
ax2.plot (x,y2, label='g2=b2', linewidth=0.5,linestyle=(0,(5,10)),color='yellow');
ax2.plot (x,s2*(x-xStar2[0])+xStar2[1],label='g2=b2+db2',linewidth=0.5, color='yellow');
ax2.axvline( 25, label='g3=b3', linewidth=0.5, color='green');
ax2.plot (x,y4, label='g4=b4', linewidth=0.5, color='red');
# DELZ
ax2.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax2.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax2.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax2.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax2.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax2.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax2.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax2.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax2.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*(fx-float(xStar1[0]))+float(xStar1[1]))
& (fy<=s2*(fx-float(xStar2[0]))+float(xStar2[1]))
& (fx<=25)
& (fy<=0*fx+20)
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax2.set_xlim(0,s);
ax2.set_ylim(0,s);
#ax2.legend();
# XSTAR
ax2.scatter(*xStar2,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&({Fraction(float(xStar2[0])).limit_denominator(10)},{Fraction(float(xStar2[1])).limit_denominator(10)})'
r'\\'
f'z^*&=&\${round(zStar2,2):,.2f}=w^*'
r'\end{eqnarray*}'
)
ax2.text(10,35,sol,fontsize=10);
# SIMPLEX
dx,dy=1,1
ax2.annotate('$A$',( 0+dx, 0+dy));
ax2.annotate('$B$',( (245/8)+dx, 0+dy));
ax2.annotate('$C$',(xStar2[0]+dx,xStar2[1]+dy));
Error in callback <function _draw_all_if_interactive at 0x1302e8e00> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/formatters.py:343, in BaseFormatter.__call__(self, obj)
341 pass
342 else:
--> 343 return printer(obj)
344 # Finally look for special method names
345 method = get_real_method(obj, self.print_method)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
<Figure size 2400x4800 with 2 Axes>
Show code cell source
d =d-rc2
# CONCISE ROW TABLEAU
rA = Matrix([
[ 'c', 'x1', 'x2', 1, 'r'],
['y1', -20, -15,b1+db1,'s1'],
['y2', -10, -20,b2+db2,'s2'],
['y3', -1, 0,b3+db3,'s3'],
['y4', 0, -1,b4+db4,'s4'],
[ -1, -500, -750, -d[0],'-z'],
[ 'c','-v1','-v2', 'w', 'r'],
])
rB = pivot(1,1,rA)
rC = pivot(2,2,rB)
rD = pivot(4,2,rC)
# CONCISE COLUMN TABLEAU
cA = Matrix([
[ 'c', 'y1', 'y2', 'y3', 'y4', 1, 'r'],
['x1', 20, 10, 1, 0,-500,'v1'],
['x2', 15, 20, 0, 1,-750,'v2'],
[ -1,b1+db1,b2+db2,b3+db3,b4+db4,d[0], 'w'],
[ 'c', '-s1', '-s2', '-s3', '-s4','-z', 'r'],
])
cB = pivot(1,1,cA)
cC = pivot(2,2,cB)
cD = pivot(2,4,cC)
display(Math(
r'\begin{aligned}'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rA = {latex(rA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rB = {latex(rB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rC = {latex(rC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rD = {latex(rD)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cA = {latex(cA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cB = {latex(cB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cC = {latex(cC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cD = {latex(cD)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\'
r'\end{aligned}'
))
\[\begin{split}\displaystyle \begin{aligned}db &= (225/2,156.25,0,0)\,\,\,rA = \left[\begin{matrix}c & x_{1} & x_{2} & 1 & r\\y_{1} & -20 & -15 & \frac{1225}{2} & s_{1}\\y_{2} & -10 & -20 & 556.25 & s_{2}\\y_{3} & -1 & 0 & 25 & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & -500 & -750 & 4528.125 & - z\\c & - v_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (225/2,156.25,0,0)\,\,\,rB = \left[\begin{matrix}c & s_{1} & x_{2} & 1 & r\\v_{1} & - \frac{1}{20} & - \frac{3}{4} & \frac{245}{8} & x_{1}\\y_{2} & \frac{1}{2} & - \frac{25}{2} & 250.0 & s_{2}\\y_{3} & \frac{1}{20} & \frac{3}{4} & - \frac{45}{8} & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & 25 & -375 & -10784.375 & - z\\c & - y_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (225/2,156.25,0,0)\,\,\,rC = \left[\begin{matrix}c & s_{1} & s_{2} & 1 & r\\v_{1} & - \frac{2}{25} & \frac{3}{50} & 15.625 & x_{1}\\v_{2} & \frac{1}{25} & - \frac{2}{25} & 20.0 & x_{2}\\y_{3} & \frac{2}{25} & - \frac{3}{50} & 9.375 & s_{3}\\y_{4} & - \frac{1}{25} & \frac{2}{25} & 0 & s_{4}\\-1 & 10 & 30 & -18284.375 & - z\\c & - y_{1} & - y_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (225/2,156.25,0,0)\,\,\,rD = \left[\begin{matrix}c & s_{1} & s_{4} & 1 & r\\v_{1} & - \frac{1}{20} & \frac{3}{4} & 15.625 & x_{1}\\v_{2} & 0 & -1 & 20.0 & x_{2}\\y_{3} & \frac{1}{20} & - \frac{3}{4} & 9.375 & s_{3}\\y_{2} & \frac{1}{2} & \frac{25}{2} & 0 & s_{2}\\-1 & 25 & 375 & -18284.375 & - z\\c & - y_{1} & - y_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\db &= (225/2,156.25,0,0)\,\,\,cA = \left[\begin{matrix}c & y_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\x_{1} & 20 & 10 & 1 & 0 & -500 & v_{1}\\x_{2} & 15 & 20 & 0 & 1 & -750 & v_{2}\\-1 & \frac{1225}{2} & 556.25 & 25 & 20 & -4528.125 & w\\c & - s_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (225/2,156.25,0,0)\,\,\,cB = \left[\begin{matrix}c & v_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{1}{20} & - \frac{1}{2} & - \frac{1}{20} & 0 & 25 & y_{1}\\x_{2} & \frac{3}{4} & \frac{25}{2} & - \frac{3}{4} & 1 & -375 & v_{2}\\-1 & \frac{245}{8} & 250.0 & - \frac{45}{8} & 20 & 10784.375 & w\\c & - x_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (225/2,156.25,0,0)\,\,\,cC = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{2}{25} & - \frac{1}{25} & - \frac{2}{25} & \frac{1}{25} & 10 & y_{1}\\s_{2} & - \frac{3}{50} & \frac{2}{25} & \frac{3}{50} & - \frac{2}{25} & 30 & y_{2}\\-1 & 15.625 & 20.0 & 9.375 & 0 & 18284.375 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (225/2,156.25,0,0)\,\,\,cD = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{2} & 1 & r\\s_{1} & \frac{1}{20} & 0 & - \frac{1}{20} & - \frac{1}{2} & 25 & y_{1}\\s_{4} & - \frac{3}{4} & 1 & \frac{3}{4} & - \frac{25}{2} & 375 & y_{4}\\-1 & 15.625 & 20.0 & 9.375 & 0 & 18284.375 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{2} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\\end{aligned}\end{split}\]
Show code cell source
ycStar2=yStar
ydStar2=(25,0,0,375)
ROI3c=round((ycStar2[0]-p1)/p1,2)
ROI4c=round((ycStar2[1]-p2)/p2,2)
ROI3d=round((ydStar2[0]-p1)/p1,2)
ROI4d=round((ydStar2[1]-p2)/p2,2)
display(Math(
r"\begin{aligned}"
r"x^*&="
f"({Fraction(float(xStar2[0])).limit_denominator(10)},{Fraction(float(xStar2[1])).limit_denominator(10)})"
r"\\"
r"z^*&="
f"\${zStar2:,.2f}"
f"&&={latex(z-rc2-rc)}"
r"=w^*\\"
r"yc^*&="
f"{ycStar2}"
r"\\"
r"\text{ROI}_\text{material}&="
f"{ROI3c}"
r"&&=\frac{yc_1^*-p_1}{p_1}"
r"\\"
r"\text{ROI}_\text{labor}&="
f"{ROI4c}"
r"&&=\frac{yc_2^*-p_2}{p_2}"
r"\\"
r"yd^*&="
f"{ydStar2}"
r"\\"
r"\text{ROI}_\text{material}&="
f"{ROI3d}"
r"&&=\frac{yd_1^*-p_1}{p_1}"
r"\\"
r"\text{ROI}_\text{labor}&="
f"{ROI4d}"
r"&&=\frac{yd_2^*-p_2}{p_2}"
r"\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}x^*&=(125/8,20)\\z^*&=\$18,284.38&&=500 x_{1} + 750 x_{2} - 4528.125=w^*\\yc^*&=(10, 30, 0, 0)\\\text{ROI}_\text{material}&=0.11&&=\frac{yc_1^*-p_1}{p_1}\\\text{ROI}_\text{labor}&=0.33&&=\frac{yc_2^*-p_2}{p_2}\\yd^*&=(25, 0, 0, 375)\\\text{ROI}_\text{material}&=1.78&&=\frac{yd_1^*-p_1}{p_1}\\\text{ROI}_\text{labor}&=-1.0&&=\frac{yd_2^*-p_2}{p_2}\\\end{aligned}\end{split}\]
Show code cell source
fig=plt.figure(dpi=300,figsize=(8,16));
##########
ax =plt.subplot(121);
ax.set_aspect(1);
# LINES
ax.plot (x,y1, label='g1=b1', linewidth=0.5,linestyle=(0,(5,10)),color='blue');
ax.plot (x,s1*(x-xStar1[0])+xStar1[1],label='g1=b1+db1',linewidth=0.5, color='blue');
ax.plot (x,y2, label='g2=b2', linewidth=0.5, color='yellow');
ax.axvline( 25, label='g3=b3', linewidth=0.5, color='green');
ax.plot (x,y4, label='g4=b4', linewidth=0.5, color='red');
# DELZ
ax.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*(fx-float(xStar1[0]))+float(xStar1[1]))
& (fy<=s2*fx+20)
& (fx<=25)
& (fy<= 0*fx+20)
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax.set_xlim(0,s);
ax.set_ylim(0,s);
#ax.legend();
# XSTAR
ax.scatter(*xStar1,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&{xStar1}'
r'\\'
f'z^*&=&\${round(zStar1,2):,.2f}=w^*'
r'\\'
f'yc^*&=&{ycStar1}'
r'\\'
f'yd^*&=&{ydStar1}'
r'\\'
r'\mathrm{ROI}_\mathrm{material}'
f'&=&{ROI1d}'
r'\\'
r'\mathrm{ROI}_\mathrm{labor}'
f'&=&{ROI2d}'
r'\end{eqnarray*}'
)
ax.text(10,35,sol,fontsize=10);
# SIMPLEX
dx,dy=1,1
ax.annotate('$A$',( 0+dx, 0+dy));
ax.annotate('$B$',( (245/8)+dx, 0+dy));
ax.annotate('$C$',(xStar1[0]+dx,xStar1[1]+dy));
##########
ax2 =plt.subplot(122);
ax2.set_aspect(1);
# LINES
ax2.plot (x,y1, label='g1=b1', linewidth=0.5,linestyle=(0,(5,10)),color='blue');
ax2.plot (x,s1*(x-xStar1[0])+xStar1[1],label='g1=b1+db1',linewidth=0.5, color='blue');
ax2.plot (x,y2, label='g2=b2', linewidth=0.5,linestyle=(0,(5,10)),color='yellow');
ax2.plot (x,s2*(x-xStar2[0])+xStar2[1],label='g2=b2+db2',linewidth=0.5, color='yellow');
ax2.axvline( 25, label='g3=b3', linewidth=0.5, color='green');
ax2.plot (x,y4, label='g4=b4', linewidth=0.5, color='red');
# DELZ
ax2.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax2.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax2.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax2.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax2.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax2.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax2.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax2.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax2.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*(fx-float(xStar1[0]))+float(xStar1[1]))
& (fy<=s2*(fx-float(xStar2[0]))+float(xStar2[1]))
& (fx<=25)
& (fy<=0*fx+20)
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax2.set_xlim(0,s);
ax2.set_ylim(0,s);
#ax2.legend();
# XSTAR
ax2.scatter(*xStar2,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&({Fraction(float(xStar2[0])).limit_denominator(10)},{Fraction(float(xStar2[1])).limit_denominator(10)})'
r'\\'
f'z^*&=&\${round(zStar2,2):,.2f}=w^*'
r'\\'
f'yc^*&=&{ycStar2}'
r'\\'
f'yd^*&=&{ydStar2}'
r'\\'
r'\mathrm{ROI}_\mathrm{material}'
f'&=&{ROI3d}'
r'\\'
r'\mathrm{ROI}_\mathrm{labor}'
f'&=&{ROI4d}'
r'\end{eqnarray*}'
)
ax2.text(10,35,sol,fontsize=10);
# SIMPLEX
dx,dy=1,1
ax2.annotate('$A$',( 0+dx, 0+dy));
ax2.annotate('$B$',( (245/8)+dx, 0+dy));
ax2.annotate('$C$',(xStar2[0]+dx,xStar2[1]+dy));
Error in callback <function _draw_all_if_interactive at 0x1302e8e00> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/formatters.py:343, in BaseFormatter.__call__(self, obj)
341 pass
342 else:
--> 343 return printer(obj)
344 # Finally look for special method names
345 method = get_real_method(obj, self.print_method)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
<Figure size 2400x4800 with 2 Axes>
[D] How much additional capital does the company need to maximize profit?#
Show code cell source
xStar4=(25,20)
db1 =g1.subs([(x1,xStar4[0]),(x2,xStar4[1])])
db2 =g2.subs([(x1,xStar4[0]),(x2,xStar4[1])])
rc4 =p1*db1+p2*db2
zStar4=z.subs([(x1,xStar4[0]),(x2,xStar4[1])])
display(Math(
r"\begin{aligned}"
f"g_1(25,20)&={latex(db1+b1)}"
r"\,\,\,\text{hrs total}"
r"\\"
f"g_2(25,20)&={latex(g2.subs([(x1,25),(x2,20)])+b2)}"
r"\,\,\,\text{lbs total}"
r"\\"
f"g_1(25,20)-b_1&={latex(db1)}"
r"\,\,\,\text{hrs additional}"
r"\\"
f"g_2(25,20)-b_2&={latex(db2)}"
r"\,\,\,\text{lbs additional}"
r"\\"
r"\text{cost of additional material and labor}&="
f"\${rc4:,.2f}"
r"\\"
r"z^*"
f"{xStar4}"
r"&="
f"\${zStar4-rc4:,.2f}"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}g_1(25,20)&=800\,\,\,\text{hrs total}\\g_2(25,20)&=650\,\,\,\text{lbs total}\\g_1(25,20)-b_1&=300\,\,\,\text{hrs additional}\\g_2(25,20)-b_2&=250\,\,\,\text{lbs additional}\\\text{cost of additional material and labor}&=\$8,325.00\\z^*(25, 20)&=\$19,175.00\end{aligned}\end{split}\]
Show code cell source
d =np.array([-rc4])
# CONCISE ROW TABLEAU
rA = Matrix([
[ 'c', 'x1', 'x2', 1, 'r'],
['y1', -20, -15,b1+db1,'s1'],
['y2', -10, -20,b2+db2,'s2'],
['y3', -1, 0,b3+db3,'s3'],
['y4', 0, -1,b4+db4,'s4'],
[ -1, -500, -750, -d[0],'-z'],
[ 'c','-v1', '-v2', 'w', 'r'],
])
rB = pivot(1,1,rA)
rC = pivot(2,2,rB)
rD = pivot(4,2,rC)
rE = pivot(3,1,rD)
# CONCISE COLUMN TABLEAU
cA = Matrix([
[ 'c', 'y1', 'y2', 'y3', 'y4', 1, 'r'],
['x1', 20, 10, 1, 0,-500,'v1'],
['x2', 15, 20, 0, 1,-750,'v2'],
[ -1,b1+db1,b2+db2,b3+db3,b4+db4,d[0], 'w'],
[ 'c', '-s1', '-s2', '-s3', '-s4','-z', 'r'],
])
cB = pivot(1,1,cA)
cC = pivot(2,2,cB)
cD = pivot(2,4,cC)
cE = pivot(1,3,cD)
display(Math(
r'\begin{aligned}'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rA = {latex(rA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rB = {latex(rB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rC = {latex(rC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rD = {latex(rD)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rE = {latex(rE)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_3=s_4=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cA = {latex(cA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cB = {latex(cB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cC = {latex(cC)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cD = {latex(cD)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cE = {latex(cE)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{LEVEL SET}\\&s_3=s_4=0\end{aligned}\\'
r'\end{aligned}'
))
\[\begin{split}\displaystyle \begin{aligned}db &= (300,250,0,0)\,\,\,rA = \left[\begin{matrix}c & x_{1} & x_{2} & 1 & r\\y_{1} & -20 & -15 & 800 & s_{1}\\y_{2} & -10 & -20 & 650 & s_{2}\\y_{3} & -1 & 0 & 25 & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & -500 & -750 & 8325.0 & - z\\c & - v_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (300,250,0,0)\,\,\,rB = \left[\begin{matrix}c & s_{1} & x_{2} & 1 & r\\v_{1} & - \frac{1}{20} & - \frac{3}{4} & 40 & x_{1}\\y_{2} & \frac{1}{2} & - \frac{25}{2} & 250 & s_{2}\\y_{3} & \frac{1}{20} & \frac{3}{4} & -15 & s_{3}\\y_{4} & 0 & -1 & 20 & s_{4}\\-1 & 25 & -375 & -11675.0 & - z\\c & - y_{1} & - v_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (300,250,0,0)\,\,\,rC = \left[\begin{matrix}c & s_{1} & s_{2} & 1 & r\\v_{1} & - \frac{2}{25} & \frac{3}{50} & 25 & x_{1}\\v_{2} & \frac{1}{25} & - \frac{2}{25} & 20 & x_{2}\\y_{3} & \frac{2}{25} & - \frac{3}{50} & 0 & s_{3}\\y_{4} & - \frac{1}{25} & \frac{2}{25} & 0 & s_{4}\\-1 & 10 & 30 & -19175.0 & - z\\c & - y_{1} & - y_{2} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (300,250,0,0)\,\,\,rD = \left[\begin{matrix}c & s_{1} & s_{4} & 1 & r\\v_{1} & - \frac{1}{20} & \frac{3}{4} & 25 & x_{1}\\v_{2} & 0 & -1 & 20 & x_{2}\\y_{3} & \frac{1}{20} & - \frac{3}{4} & 0 & s_{3}\\y_{2} & \frac{1}{2} & \frac{25}{2} & 0 & s_{2}\\-1 & 25 & 375 & -19175.0 & - z\\c & - y_{1} & - y_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\db &= (300,250,0,0)\,\,\,rE = \left[\begin{matrix}c & s_{3} & s_{4} & 1 & r\\v_{1} & -1 & 0 & 25 & x_{1}\\v_{2} & 0 & -1 & 20 & x_{2}\\y_{1} & 20 & 15 & 0 & s_{1}\\y_{2} & 10 & 20 & 0 & s_{2}\\-1 & 500 & 750 & -19175.0 & - z\\c & - y_{3} & - y_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_3=s_4=0\end{aligned}\\db &= (300,250,0,0)\,\,\,cA = \left[\begin{matrix}c & y_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\x_{1} & 20 & 10 & 1 & 0 & -500 & v_{1}\\x_{2} & 15 & 20 & 0 & 1 & -750 & v_{2}\\-1 & 800 & 650 & 25 & 20 & -8325.0 & w\\c & - s_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (300,250,0,0)\,\,\,cB = \left[\begin{matrix}c & v_{1} & y_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{1}{20} & - \frac{1}{2} & - \frac{1}{20} & 0 & 25 & y_{1}\\x_{2} & \frac{3}{4} & \frac{25}{2} & - \frac{3}{4} & 1 & -375 & v_{2}\\-1 & 40 & 250 & -15 & 20 & 11675.0 & w\\c & - x_{1} & - s_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (300,250,0,0)\,\,\,cC = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{4} & 1 & r\\s_{1} & \frac{2}{25} & - \frac{1}{25} & - \frac{2}{25} & \frac{1}{25} & 10 & y_{1}\\s_{2} & - \frac{3}{50} & \frac{2}{25} & \frac{3}{50} & - \frac{2}{25} & 30 & y_{2}\\-1 & 25 & 20 & 0 & 0 & 19175.0 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&s_1=s_2=0\end{aligned}\\db &= (300,250,0,0)\,\,\,cD = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{2} & 1 & r\\s_{1} & \frac{1}{20} & 0 & - \frac{1}{20} & - \frac{1}{2} & 25 & y_{1}\\s_{4} & - \frac{3}{4} & 1 & \frac{3}{4} & - \frac{25}{2} & 375 & y_{4}\\-1 & 25 & 20 & 0 & 0 & 19175.0 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{2} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_1=s_4=0\end{aligned}\\db &= (300,250,0,0)\,\,\,cE = \left[\begin{matrix}c & v_{1} & v_{2} & y_{1} & y_{2} & 1 & r\\s_{3} & 1 & 0 & -20 & -10 & 500 & y_{3}\\s_{4} & 0 & 1 & -15 & -20 & 750 & y_{4}\\-1 & 25 & 20 & 0 & 0 & 19175.0 & w\\c & - x_{1} & - x_{2} & - s_{1} & - s_{2} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{LEVEL SET}\\&s_3=s_4=0\end{aligned}\\\end{aligned}\end{split}\]
ycStar4=yStar
ydStar4=(25,0,0,375)
yeStar4=(0,0,500,750)
ROI5c=round((ycStar4[0]-p1)/p1,2)
ROI6c=round((ycStar4[1]-p2)/p2,2)
ROI5d=round((ydStar4[0]-p1)/p1,2)
ROI6d=round((ydStar4[1]-p2)/p2,2)
ROI5e=round((yeStar4[0]-p1)/p1,2)
ROI6e=round((yeStar4[1]-p2)/p2,2)
Show code cell source
fig=plt.figure(dpi=300,figsize=(8,16));
##########
ax2 =plt.subplot();
ax2.set_aspect(1);
# LINES
ax2.plot (x,y1, label='g1=b1', linewidth=0.8,linestyle=(0,(5,10)),color='blue');
ax2.plot (x,s1*(x-xStar4[0])+xStar4[1],label='g1=b1+db1',linewidth=0.8, color='blue');
ax2.plot (x,y2, label='g2=b2', linewidth=0.8,linestyle=(0,(5,10)),color='yellow');
ax2.plot (x,s2*(x-xStar4[0])+xStar4[1],label='g2=b2+db2',linewidth=0.8, color='yellow');
ax2.axvline( 25, label='g3=b3', linewidth=0.5, color='green');
ax2.plot (x,y4, label='g4=b4', linewidth=0.5, color='red');
# DELZ
ax2.arrow(10,10,*unitVector( delz),width=0.01,head_width=0.2,length_includes_head=True,color='black');
# FEASIBLE SIDES <=
ax2.arrow(10,10,*unitVector(-delg1),width=0.01,head_width=0.2,length_includes_head=True,color='blue');
ax2.arrow(10,10,*unitVector(-delg2),width=0.01,head_width=0.2,length_includes_head=True,color='yellow');
ax2.arrow(10,10,*unitVector(-delg3),width=0.01,head_width=0.2,length_includes_head=True,color='green');
ax2.arrow(10,10,*unitVector(-delg4),width=0.01,head_width=0.2,length_includes_head=True,color='red');
# DIRECTIONS
ax2.arrow(25, 0,*udg1*3, width=0.3, head_width=1, length_includes_head=True,color='blue');
ax2.arrow( 0,20,*udg2*3, width=0.3, head_width=1, length_includes_head=True,color='yellow');
ax2.arrow(25, 0,*udg3*3, width=0.3, head_width=1, length_includes_head=True,color='green');
ax2.arrow( 0,20,*udg4*3, width=0.3, head_width=1, length_includes_head=True,color='red');
# FEASIBLE REGION
fx,fy=np.meshgrid(x,x)
plt.imshow(
(
(fy<=s1*(fx-float(xStar4[0]))+float(xStar4[1]))
& (fy<=s2*(fx-float(xStar4[0]))+float(xStar4[1]))
& (fx<=25)
& (fy<=0*fx+20)
).astype(int),
extent=(fx.min(),fx.max(),fy.min(),fy.max()),
origin='lower',
cmap ='Greys',
alpha =0.3,
);
ax2.set_xlim(0,s);
ax2.set_ylim(0,s);
ax2.legend();
# XSTAR
ax2.scatter(*xStar4,s=1e2,color='purple');
sol=(
r'\begin{eqnarray*}'
f'x^*&=&({Fraction(float(xStar4[0])).limit_denominator(10)},{Fraction(float(xStar4[1])).limit_denominator(10)})'
r'\\'
f'z^*&=&\${round(zStar4-rc4,2):,.2f}=w^*'
r'\\'
f'yc^*&=&{ycStar4}'
r'\\'
f'yd^*&=&{ydStar4}'
r'\\'
r'\mathrm{ROI}_\mathrm{material}'
f'&=&{ROI5e}'
r'\\'
r'\mathrm{ROI}_\mathrm{labor}'
f'&=&{ROI6e}'
r'\end{eqnarray*}'
)
ax2.text(10,35,sol,fontsize=10);
# SIMPLEX
dx,dy=1,1
ax2.annotate('$A$',( 0+dx, 0+dy));
ax2.annotate('$B$',( 40+dx, 0+dy));
ax2.annotate('$C$',(xStar4[0]+dx,xStar4[1]+dy));
Error in callback <function _draw_all_if_interactive at 0x1302e8e00> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
---------------------------------------------------------------------------
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/formatters.py:343, in BaseFormatter.__call__(self, obj)
341 pass
342 else:
--> 343 return printer(obj)
344 # Finally look for special method names
345 method = get_real_method(obj, self.print_method)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axes/_base.py:3070, in _AxesBase.draw(self, renderer)
3067 if artists_rasterized:
3068 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3070 mimage._draw_list_compositing_images(
3071 renderer, self, artists, self.figure.suppressComposite)
3073 renderer.close_group('axes')
3074 self.stale = False
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1388, in Axis.draw(self, renderer, *args, **kwargs)
1385 renderer.open_group(__name__, gid=self.get_gid())
1387 ticks_to_draw = self._update_ticks()
-> 1388 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1390 for tick in ticks_to_draw:
1391 tick.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/axis.py:1315, in <listcomp>(.0)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
<Figure size 2400x4800 with 1 Axes>
[E] If the company only has $5,000.00 in capital for the purposes of increasing production (and not the amount determined to meet demand in [D]), what allocation of overtime and additional material maximizes the company’s profit?#
Show code cell source
display(Math(
r"\begin{aligned}"
r"\text{ROI}_\text{material}&="
f"{ROI1}"
r"&&\text{return on investment per dollar invested in material}"
r"\\"
r"\text{ROI}_\text{labor}&="
f"{ROI2}"
r"&&\text{return on investment per dollar invested in labor}"
r"\\"
r"\end{aligned}"
))
\[\begin{split}\displaystyle \begin{aligned}\text{ROI}_\text{material}&=0.11&&\text{return on investment per dollar invested in material}\\\text{ROI}_\text{labor}&=0.33&&\text{return on investment per dollar invested in labor}\\\end{aligned}\end{split}\]
Both purchasing additional material and scheduling overtime are profitable, but the limited budget constrains us to make certain considerations.
Scheduling overtime is more profitable than purchasing additional material since the return on its investment is higher.
Make the additional material purchased [db1=x3] and overtime schedule [db2=x4] new control variables.
\( \begin{aligned} g_i\le b_i+db_i &\iff g_i-db_i\le b_i \\ g_1\le b_1+db_1 &\iff g_1-db_1\le b_1 \implies g_1-x_3\le b_1 \\ g_2\le b_2+db_2 &\iff g_2-db_2\le b_2 \implies g_2-x_4\le b_2 \end{aligned} \)
\( \begin{aligned} \max z(\mathbf{x})&=500x_1+750x_2-(9.00x_3+22.50x_4) \\ g_1(\mathbf{x})&=20x_1+15x_2-x_3\le500=b_1 &&\text{material constraint} \\ g_2(\mathbf{x})&=10x_1+20x_2-x_4\le400=b_2 &&\text{labor constraint} \\ g_3(\mathbf{x})&=x_1\le25=b_3 &&\text{Type A valve demand constaint} \\ g_4(\mathbf{x})&=x_2\le20=b_4 &&\text{Type B valve demand constraint} \\ g_5(\mathbf{x})&=9.00x_3+22.50x_4\le5000=b_5 &&\text{budget constraint} \\ \mathbf{x}&\ge\mathbf{0} &&\text{no negative production} \end{aligned} \)
LP A B
max z = 500x1 + 750x2 - (9.00x3 + 22.50x4)
g1 = 20x1 + 15x2 - 1x3 + 0x4 <= 500 = b1 material constraint
g2 = 10x1 + 20x2 0x3 - 1x4 <= 400 = b2 labor constraint
g3 = 1x1 + 0x2 + 0x3 + 0x4 <= 25 = b3 Type A valve demand constraint
g4 = 0x1 + 1x2 + 0x3 + 0x4 <= 20 = b4 Type B valve demand constraint
g5 = 0x1 + 0x2 + 9.00x3 + 22.50x4 <= 5000 = b5 budget constraint
x >= 0 no negative production
d = 0
b1= 500
b2= 400
b3= 25
b4= 20
b5=5000
z =500*x1+750*x2-9*x3-(45/2)*x4- d
g1= 20*x1+ 15*x2 -x3 -b1
g2= 10*x1+ 20*x2 -x4-b2
g3= 1*x1+ 0*x2 -b3
g4= 0*x1+ 1*x2 -b4
g5= 9*x3+(45/2)*x4-b5
db1=0
db2=0
db3=0
db4=0
db5=0
A=np.array([[20,15,-1,0],[10,20,0,-1],[1,0,0,0],[0,1,0,0],[0,0,9,(45/2)]])
b=np.array([500,400,25,20,5000])
c=np.array([500,750,-9,-(45/2)])
d=np.array([0])
x=np.array([x1,x2,x3,x4])
display(Math(
r"\begin{aligned}"
fr" z&={latex( c@x+d [0])}\\"
fr"g_1&={latex((A@x-b)[0])}\\"
fr"g_2&={latex((A@x-b)[1])}\\"
fr"g_3&={latex((A@x-b)[2])}\\"
fr"g_4&={latex((A@x-b)[3])}\\"
fr"g_5&={latex((A@x-b)[4])}\\"
r"\end{aligned}"),
)
\[\begin{split}\displaystyle \begin{aligned} z&=500.0 x_{1} + 750.0 x_{2} - 9.0 x_{3} - 22.5 x_{4}\\g_1&=20.0 x_{1} + 15.0 x_{2} - 1.0 x_{3} - 500\\g_2&=10.0 x_{1} + 20.0 x_{2} - 1.0 x_{4} - 400\\g_3&=1.0 x_{1} - 25\\g_4&=1.0 x_{2} - 20\\g_5&=9.0 x_{3} + 22.5 x_{4} - 5000\\\end{aligned}\end{split}\]
Show code cell source
# CONCISE ROW TABLEAU
rA = Matrix([
[ 'c', 'x1', 'x2', 'x3', 'x4', 1, 'r'],
['y1', -20, -15, 1, 0, b1+db1,'s1'],
['y2', -10, -20, 0, 1, b2+db2,'s2'],
['y3', -1, 0, 0, 0, b3+db3,'s3'],
['y4', 0, -1, 0, 0, b4+db4,'s4'],
['y5', 0, 0, -9,-(45/2), b5+db5,'s5'],
[ -1, -500, -750, 9, (45/2), -d[0],'-z'],
[ 'c','-v1','-v2','-v3', '-v4', 'w', 'r'],
])
rB = pivot(1,1,rA)
rC = pivot(2,2,rB)
rD = pivot(3,3,rC)
rE = pivot(5,4,rD)
rF = pivot(4,3,rE)
# CONCISE COLUMN TABLEAU
cA = Matrix([
[ 'c', 'y1', 'y2', 'y3', 'y4', 'y5', 1, 'r'],
['x1', 20, 10, 1, 0, 0, -500,'v1'],
['x2', 15, 20, 0, 1, 0, -750,'v2'],
['x3', -1, 0, 0, 0, 9, 9,'v3'],
['x4', 0, -1, 0, 0,(45/2),(45/2),'v4'],
[ -1,b1+db1,b2+db2,b3+db3,b4+db4,b5+db5, d[0], 'w'],
[ 'c', '-s1', '-s2', '-s3', '-s4', '-s5', '-z', 'r'],
])
cB = pivot(1,1,cA)
cC = pivot(2,2,cB)
cD = pivot(3,3,cC)
cE = pivot(4,5,cD)
cF = pivot(3,4,cE)
display(Math(
r'\begin{aligned}'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rA = {latex(rA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rB = {latex(rB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rC = {latex(rC)}\,\,\,\,\,&&s_1=s_2=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rD = {latex(rD)}\,\,\,\,\,&&\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rE = {latex(nsimplify(rE))}\,\,\,\,\,&&\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,rF = {latex(nsimplify(rF))}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cA = {latex(cA)}\,\,\,\,\,&&'
r'\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cB = {latex(cB)}\,\,\,\,\,&&x_2=s_1=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cC = {latex(cC)}\,\,\,\,\,&&s_1=s_2=0\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cD = {latex(cD)}\,\,\,\,\,&&\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cE = {latex(nsimplify(cE))}\,\,\,\,\,&&\\'
fr'db &= ({db1},{db2},{db3},{db4})\,\,\,cF = {latex(nsimplify(cF))}\,\,\,\,\,&&'
r'\begin{aligned}&\text{OPTIMAL}\\&\end{aligned}\\'
r'\end{aligned}'
))
\[\begin{split}\displaystyle \begin{aligned}db &= (0,0,0,0)\,\,\,rA = \left[\begin{matrix}c & x_{1} & x_{2} & x_{3} & x_{4} & 1 & r\\y_{1} & -20 & -15 & 1 & 0 & 500 & s_{1}\\y_{2} & -10 & -20 & 0 & 1 & 400 & s_{2}\\y_{3} & -1 & 0 & 0 & 0 & 25 & s_{3}\\y_{4} & 0 & -1 & 0 & 0 & 20 & s_{4}\\y_{5} & 0 & 0 & -9 & -22.5 & 5000 & s_{5}\\-1 & -500 & -750 & 9 & 22.5 & 0 & - z\\c & - v_{1} & - v_{2} & - v_{3} & - v_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (0,0,0,0)\,\,\,rB = \left[\begin{matrix}c & s_{1} & x_{2} & x_{3} & x_{4} & 1 & r\\v_{1} & - \frac{1}{20} & - \frac{3}{4} & \frac{1}{20} & 0 & 25 & x_{1}\\y_{2} & \frac{1}{2} & - \frac{25}{2} & - \frac{1}{2} & 1 & 150 & s_{2}\\y_{3} & \frac{1}{20} & \frac{3}{4} & - \frac{1}{20} & 0 & 0 & s_{3}\\y_{4} & 0 & -1 & 0 & 0 & 20 & s_{4}\\y_{5} & 0 & 0 & -9 & -22.5 & 5000 & s_{5}\\-1 & 25 & -375 & -16 & 22.5 & -12500 & - z\\c & - y_{1} & - v_{2} & - v_{3} & - v_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (0,0,0,0)\,\,\,rC = \left[\begin{matrix}c & s_{1} & s_{2} & x_{3} & x_{4} & 1 & r\\v_{1} & - \frac{2}{25} & \frac{3}{50} & \frac{2}{25} & - \frac{3}{50} & 16 & x_{1}\\v_{2} & \frac{1}{25} & - \frac{2}{25} & - \frac{1}{25} & \frac{2}{25} & 12 & x_{2}\\y_{3} & \frac{2}{25} & - \frac{3}{50} & - \frac{2}{25} & \frac{3}{50} & 9 & s_{3}\\y_{4} & - \frac{1}{25} & \frac{2}{25} & \frac{1}{25} & - \frac{2}{25} & 8 & s_{4}\\y_{5} & 0 & 0 & -9 & -22.5 & 5000 & s_{5}\\-1 & 10 & 30 & -1 & -7.5 & -17000 & - z\\c & - y_{1} & - y_{2} & - v_{3} & - v_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&s_1=s_2=0\\db &= (0,0,0,0)\,\,\,rD = \left[\begin{matrix}c & s_{1} & s_{2} & s_{3} & x_{4} & 1 & r\\v_{1} & 0 & 0 & -1 & 0 & 25 & x_{1}\\v_{2} & 0 & - \frac{1}{20} & \frac{1}{2} & \frac{1}{20} & \frac{15}{2} & x_{2}\\v_{3} & 1 & - \frac{3}{4} & - \frac{25}{2} & \frac{3}{4} & \frac{225}{2} & x_{3}\\y_{4} & 0 & \frac{1}{20} & - \frac{1}{2} & - \frac{1}{20} & \frac{25}{2} & s_{4}\\y_{5} & -9 & \frac{27}{4} & \frac{225}{2} & -29.25 & \frac{7975}{2} & s_{5}\\-1 & 9 & \frac{123}{4} & \frac{25}{2} & -8.25 & - \frac{34225}{2} & - z\\c & - y_{1} & - y_{2} & - y_{3} & - v_{4} & w & r\end{matrix}\right]\,\,\,\,\,&&\\db &= (0,0,0,0)\,\,\,rE = \left[\begin{matrix}c & s_{1} & s_{2} & s_{3} & s_{5} & 1 & r\\v_{1} & 0 & 0 & -1 & 0 & 25 & x_{1}\\v_{2} & - \frac{1}{65} & - \frac{1}{26} & \frac{9}{13} & - \frac{1}{585} & \frac{143162393162393}{10000000000000} & x_{2}\\v_{3} & \frac{10}{13} & - \frac{15}{26} & - \frac{125}{13} & - \frac{1}{39} & \frac{21474358974359}{100000000000} & x_{3}\\y_{4} & \frac{1}{65} & \frac{1}{26} & - \frac{9}{13} & \frac{1}{585} & \frac{142094017094017}{25000000000000} & s_{4}\\v_{4} & - \frac{4}{13} & \frac{3}{13} & \frac{50}{13} & - \frac{4}{117} & \frac{68162393162393}{500000000000} & x_{4}\\-1 & \frac{150}{13} & \frac{375}{13} & - \frac{250}{13} & \frac{11}{39} & - \frac{36474358974359}{2000000000} & - z\\c & - y_{1} & - y_{2} & - y_{3} & - y_{5} & w & r\end{matrix}\right]\,\,\,\,\,&&\\db &= (0,0,0,0)\,\,\,rF = \left[\begin{matrix}c & s_{1} & s_{2} & s_{4} & s_{5} & 1 & r\\v_{1} & - \frac{1}{45} & - \frac{1}{18} & \frac{13}{9} & - \frac{1}{405} & \frac{167901234567901}{10000000000000} & x_{1}\\v_{2} & 0 & 0 & -1 & - \frac{216840434497101}{1000000000000000000000000000000000} & 20 & x_{2}\\v_{3} & \frac{5}{9} & - \frac{10}{9} & \frac{125}{9} & - \frac{4}{81} & \frac{67901234567901}{500000000000} & x_{3}\\y_{3} & \frac{1}{45} & \frac{1}{18} & - \frac{13}{9} & \frac{1}{405} & \frac{665}{81} & s_{3}\\v_{4} & - \frac{2}{9} & \frac{4}{9} & - \frac{50}{9} & - \frac{2}{81} & \frac{167901234567901}{1000000000000} & x_{4}\\-1 & \frac{100}{9} & \frac{250}{9} & \frac{250}{9} & \frac{19}{81} & - \frac{183950617283951}{10000000000} & - z\\c & - y_{1} & - y_{2} & - y_{4} & - y_{5} & w & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&\end{aligned}\\db &= (0,0,0,0)\,\,\,cA = \left[\begin{matrix}c & y_{1} & y_{2} & y_{3} & y_{4} & y_{5} & 1 & r\\x_{1} & 20 & 10 & 1 & 0 & 0 & -500 & v_{1}\\x_{2} & 15 & 20 & 0 & 1 & 0 & -750 & v_{2}\\x_{3} & -1 & 0 & 0 & 0 & 9 & 9 & v_{3}\\x_{4} & 0 & -1 & 0 & 0 & 22.5 & 22.5 & v_{4}\\-1 & 500 & 400 & 25 & 20 & 5000 & 0 & w\\c & - s_{1} & - s_{2} & - s_{3} & - s_{4} & - s_{5} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{START}\\&x_1=x_2=0\end{aligned}\\db &= (0,0,0,0)\,\,\,cB = \left[\begin{matrix}c & v_{1} & y_{2} & y_{3} & y_{4} & y_{5} & 1 & r\\s_{1} & \frac{1}{20} & - \frac{1}{2} & - \frac{1}{20} & 0 & 0 & 25 & y_{1}\\x_{2} & \frac{3}{4} & \frac{25}{2} & - \frac{3}{4} & 1 & 0 & -375 & v_{2}\\x_{3} & - \frac{1}{20} & \frac{1}{2} & \frac{1}{20} & 0 & 9 & -16 & v_{3}\\x_{4} & 0 & -1 & 0 & 0 & 22.5 & 22.5 & v_{4}\\-1 & 25 & 150 & 0 & 20 & 5000 & 12500 & w\\c & - x_{1} & - s_{2} & - s_{3} & - s_{4} & - s_{5} & - z & r\end{matrix}\right]\,\,\,\,\,&&x_2=s_1=0\\db &= (0,0,0,0)\,\,\,cC = \left[\begin{matrix}c & v_{1} & v_{2} & y_{3} & y_{4} & y_{5} & 1 & r\\s_{1} & \frac{2}{25} & - \frac{1}{25} & - \frac{2}{25} & \frac{1}{25} & 0 & 10 & y_{1}\\s_{2} & - \frac{3}{50} & \frac{2}{25} & \frac{3}{50} & - \frac{2}{25} & 0 & 30 & y_{2}\\x_{3} & - \frac{2}{25} & \frac{1}{25} & \frac{2}{25} & - \frac{1}{25} & 9 & -1 & v_{3}\\x_{4} & \frac{3}{50} & - \frac{2}{25} & - \frac{3}{50} & \frac{2}{25} & 22.5 & -7.5 & v_{4}\\-1 & 16 & 12 & 9 & 8 & 5000 & 17000 & w\\c & - x_{1} & - x_{2} & - s_{3} & - s_{4} & - s_{5} & - z & r\end{matrix}\right]\,\,\,\,\,&&s_1=s_2=0\\db &= (0,0,0,0)\,\,\,cD = \left[\begin{matrix}c & v_{1} & v_{2} & v_{3} & y_{4} & y_{5} & 1 & r\\s_{1} & 0 & 0 & -1 & 0 & 9 & 9 & y_{1}\\s_{2} & 0 & \frac{1}{20} & \frac{3}{4} & - \frac{1}{20} & - \frac{27}{4} & \frac{123}{4} & y_{2}\\s_{3} & 1 & - \frac{1}{2} & \frac{25}{2} & \frac{1}{2} & - \frac{225}{2} & \frac{25}{2} & y_{3}\\x_{4} & 0 & - \frac{1}{20} & - \frac{3}{4} & \frac{1}{20} & 29.25 & -8.25 & v_{4}\\-1 & 25 & \frac{15}{2} & \frac{225}{2} & \frac{25}{2} & \frac{7975}{2} & \frac{34225}{2} & w\\c & - x_{1} & - x_{2} & - x_{3} & - s_{4} & - s_{5} & - z & r\end{matrix}\right]\,\,\,\,\,&&\\db &= (0,0,0,0)\,\,\,cE = \left[\begin{matrix}c & v_{1} & v_{2} & v_{3} & y_{4} & v_{4} & 1 & r\\s_{1} & 0 & \frac{1}{65} & - \frac{10}{13} & - \frac{1}{65} & \frac{4}{13} & \frac{150}{13} & y_{1}\\s_{2} & 0 & \frac{1}{26} & \frac{15}{26} & - \frac{1}{26} & - \frac{3}{13} & \frac{375}{13} & y_{2}\\s_{3} & 1 & - \frac{9}{13} & \frac{125}{13} & \frac{9}{13} & - \frac{50}{13} & - \frac{250}{13} & y_{3}\\s_{5} & 0 & \frac{1}{585} & \frac{1}{39} & - \frac{1}{585} & \frac{4}{117} & \frac{11}{39} & y_{5}\\-1 & 25 & \frac{143162393162393}{10000000000000} & \frac{21474358974359}{100000000000} & \frac{142094017094017}{25000000000000} & \frac{68162393162393}{500000000000} & \frac{36474358974359}{2000000000} & w\\c & - x_{1} & - x_{2} & - x_{3} & - s_{4} & - x_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\\db &= (0,0,0,0)\,\,\,cF = \left[\begin{matrix}c & v_{1} & v_{2} & v_{3} & y_{3} & v_{4} & 1 & r\\s_{1} & \frac{1}{45} & 0 & - \frac{5}{9} & - \frac{1}{45} & \frac{2}{9} & \frac{100}{9} & y_{1}\\s_{2} & \frac{1}{18} & 0 & \frac{10}{9} & - \frac{1}{18} & - \frac{4}{9} & \frac{250}{9} & y_{2}\\s_{4} & - \frac{13}{9} & 1 & - \frac{125}{9} & \frac{13}{9} & \frac{50}{9} & \frac{250}{9} & y_{4}\\s_{5} & \frac{1}{405} & \frac{216840434497101}{1000000000000000000000000000000000} & \frac{4}{81} & - \frac{1}{405} & \frac{2}{81} & \frac{19}{81} & y_{5}\\-1 & \frac{167901234567901}{10000000000000} & 20 & \frac{67901234567901}{500000000000} & \frac{665}{81} & \frac{167901234567901}{1000000000000} & \frac{183950617283951}{10000000000} & w\\c & - x_{1} & - x_{2} & - x_{3} & - s_{3} & - x_{4} & - z & r\end{matrix}\right]\,\,\,\,\,&&\begin{aligned}&\text{OPTIMAL}\\&\end{aligned}\\\end{aligned}\end{split}\]
cF[-2,-2]
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/sympy/interactive/printing.py:178, in _init_ipython_printing.<locals>._print_latex_png(o)
177 try:
--> 178 return _preview_wrapper(s)
179 except RuntimeError as e:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/sympy/interactive/printing.py:89, in _init_ipython_printing.<locals>._preview_wrapper(o)
88 try:
---> 89 preview(o, output='png', viewer='BytesIO', euler=euler,
90 outputbuffer=exprbuffer, extra_preamble=extra_preamble,
91 dvioptions=dvioptions, fontsize=fontsize)
92 except Exception as e:
93 # IPython swallows exceptions
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/sympy/printing/preview.py:308, in preview(expr, output, viewer, euler, packages, filename, outputbuffer, preamble, dvioptions, outputTexFile, extra_preamble, fontsize, **latex_settings)
307 if not shutil.which('latex'):
--> 308 raise RuntimeError("latex program is not installed")
310 try:
RuntimeError: latex program is not installed
During handling of the above exception, another exception occurred:
PermissionError Traceback (most recent call last)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/core/formatters.py:347, in BaseFormatter.__call__(self, obj)
345 method = get_real_method(obj, self.print_method)
346 if method is not None:
--> 347 return method()
348 return None
349 else:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/sympy/interactive/printing.py:184, in _init_ipython_printing.<locals>._print_latex_png(o)
182 if latex_mode != 'inline':
183 s = latex(o, mode='inline', **settings)
--> 184 return _matplotlib_wrapper(s)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/sympy/interactive/printing.py:118, in _init_ipython_printing.<locals>._matplotlib_wrapper(o)
116 try:
117 try:
--> 118 return latex_to_png(o, color=forecolor, scale=scale)
119 except TypeError: # Old IPython version without color and scale
120 return latex_to_png(o)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/lib/latextools.py:105, in latex_to_png(s, encode, backend, wrap, color, scale)
103 else:
104 raise ValueError('No such backend {0}'.format(backend))
--> 105 bin_data = f(s, wrap, color, scale)
106 if encode and bin_data:
107 bin_data = encodebytes(bin_data)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/IPython/lib/latextools.py:135, in latex_to_png_mpl(s, wrap, color, scale)
133 fig.text(0, depth / height, s, fontproperties=prop, color=color)
134 backend_agg.FigureCanvasAgg(fig)
--> 135 fig.savefig(buffer, dpi=dpi, format="png", transparent=True)
136 return buffer.getvalue()
137 except (ValueError, RuntimeError, ParseFatalException):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3390, in Figure.savefig(self, fname, transparent, **kwargs)
3388 for ax in self.axes:
3389 _recursively_make_axes_transparent(stack, ax)
-> 3390 self.canvas.print_figure(fname, **kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2193, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2189 try:
2190 # _get_renderer may change the figure dpi (as vector formats
2191 # force the figure dpi to 72), so we need to set it again here.
2192 with cbook._setattr_cm(self.figure, dpi=dpi):
-> 2193 result = print_method(
2194 filename,
2195 facecolor=facecolor,
2196 edgecolor=edgecolor,
2197 orientation=orientation,
2198 bbox_inches_restore=_bbox_inches_restore,
2199 **kwargs)
2200 finally:
2201 if bbox_inches and restore_bbox:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:2043, in FigureCanvasBase._switch_canvas_and_return_print_method.<locals>.<lambda>(*args, **kwargs)
2039 optional_kws = { # Passed by print_figure for other renderers.
2040 "dpi", "facecolor", "edgecolor", "orientation",
2041 "bbox_inches_restore"}
2042 skip = optional_kws - {*inspect.signature(meth).parameters}
-> 2043 print_method = functools.wraps(meth)(lambda *args, **kwargs: meth(
2044 *args, **{k: v for k, v in kwargs.items() if k not in skip}))
2045 else: # Let third-parties do as they see fit.
2046 print_method = meth
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:497, in FigureCanvasAgg.print_png(self, filename_or_obj, metadata, pil_kwargs)
450 def print_png(self, filename_or_obj, *, metadata=None, pil_kwargs=None):
451 """
452 Write the figure to a PNG file.
453
(...)
495 *metadata*, including the default 'Software' key.
496 """
--> 497 self._print_pil(filename_or_obj, "png", pil_kwargs, metadata)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:445, in FigureCanvasAgg._print_pil(self, filename_or_obj, fmt, pil_kwargs, metadata)
440 def _print_pil(self, filename_or_obj, fmt, pil_kwargs, metadata=None):
441 """
442 Draw the canvas, then save it using `.image.imsave` (to which
443 *pil_kwargs* and *metadata* are forwarded).
444 """
--> 445 FigureCanvasAgg.draw(self)
446 mpl.image.imsave(
447 filename_or_obj, self.buffer_rgba(), format=fmt, origin="upper",
448 dpi=self.figure.dpi, metadata=metadata, pil_kwargs=pil_kwargs)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:748, in Text.draw(self, renderer)
745 renderer.open_group('text', self.get_gid())
747 with self._cm_set(text=self._get_wrapped_text()):
--> 748 bbox, info, descent = self._get_layout(renderer)
749 trans = self.get_transform()
751 # don't use self.get_position here, which refers to text
752 # position in Text:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/texmanager.py:366, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
--> 366 page, = dvi
367 # A total height (including the descent) needs to be returned.
368 return page.width, page.height + page.descent, page.descent
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:296, in Dvi.__iter__(self)
280 def __iter__(self):
281 """
282 Iterate through the pages of the file.
283
(...)
294 integers.
295 """
--> 296 while self._read():
297 yield self._output()
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:375, in Dvi._read(self)
373 while True:
374 byte = self.file.read(1)[0]
--> 375 self._dtable[byte](self, byte)
376 name = self._dtable[byte].__name__
377 if name == "_push":
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:227, in _dispatch.<locals>.decorate.<locals>.wrapper(self, byte)
225 if state is not None and self.state != state:
226 raise ValueError("state precondition failed")
--> 227 return method(self, *[f(self, byte-min) for f in get_args])
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:526, in Dvi._fnt_def(self, k, c, s, d, a, l)
524 @_dispatch(min=243, max=246, args=('olen1', 'u4', 'u4', 'u4', 'u1', 'u1'))
525 def _fnt_def(self, k, c, s, d, a, l):
--> 526 self._fnt_def_real(k, c, s, d, a, l)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:531, in Dvi._fnt_def_real(self, k, c, s, d, a, l)
529 n = self.file.read(a + l)
530 fontname = n[-l:].decode('ascii')
--> 531 tfm = _tfmfile(fontname)
532 if c != 0 and tfm.checksum != 0 and c != tfm.checksum:
533 raise ValueError('tfm checksum mismatch: %s' % n)
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1116, in _fontfile(cls, suffix, texname)
1114 @lru_cache
1115 def _fontfile(cls, suffix, texname):
-> 1116 return cls(find_tex_file(texname + suffix))
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1082, in find_tex_file(filename)
1079 filename = filename.decode('utf-8', errors='replace')
1081 try:
-> 1082 lk = _LuatexKpsewhich()
1083 except FileNotFoundError:
1084 lk = None # Fallback to directly calling kpsewhich, as below.
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1037, in _LuatexKpsewhich.__new__(cls)
1034 @lru_cache # A singleton.
1035 def __new__(cls):
1036 self = object.__new__(cls)
-> 1037 self._proc = self._new_proc()
1038 return self
File ~/anaconda3/envs/ml/lib/python3.11/site-packages/matplotlib/dviread.py:1041, in _LuatexKpsewhich._new_proc(self)
1040 def _new_proc(self):
-> 1041 return subprocess.Popen(
1042 ["luatex", "--luaonly",
1043 str(cbook._get_data_path("kpsewhich.lua"))],
1044 stdin=subprocess.PIPE, stdout=subprocess.PIPE)
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1022 if self.text_mode:
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
1037 for f in filter(None, (self.stdin, self.stdout, self.stderr)):
File ~/anaconda3/envs/ml/lib/python3.11/subprocess.py:1955, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1953 err_msg = os.strerror(errno_num)
1954 if err_filename is not None:
-> 1955 raise child_exception_type(errno_num, err_msg, err_filename)
1956 else:
1957 raise child_exception_type(errno_num, err_msg)
PermissionError: [Errno 13] Permission denied: 'luatex'
\[\displaystyle 18395.0617283951\]