Python Day 5:
**Linear Optimization problem **
Side Story: Excel Solver is an excellent tool for Linear Optimization problems, but we need to pay for this platform.
In addition, 5 asked me with an extremely surprised facial expression: "What kind of data scientists would use Excel? " 😱😱😱
Therefore we will introduce how to use Python to solve a Linear Optimization Problem, so we can all be Real data Scientists.
Problem:
Pirelli glass produces two types of fiber optic strands: A-type & H-type (a higher speed variety).
A-type H-type
Fiber 1 1
Labor 9 hours 6 hours
Glass 12 lbs 16 lbs
Unit Profit 350usd 300usd
Due to an unusual agreement they made as part of a takeover of a small company next week they are only allowed to produce 200 fibers, there are 1566 hours of labor available, and 2880 lbs of glass available.
Question:
We want to determine the best combination of A-type and H-type fibers to produce?
[caption id="attachment_1564" align="alignnone" width="750"]
image
pixel2013 / Pixabay[/caption]
Solution Key:
X1=number of A-type fibers to produce
X2=number of H-type fibers to produce
MAX Revenue :
Constrain:
fibers
labor
glass
Python Code:
from scipy.optimize import linprog
import numpy as np
import matplotlib.pyplot as plt
from numpy.linalg import solve
%matplotlib inline
# Solve
#Note linprog is for min, in order to solve for max,
#we switch the sign
c=[-350, -300]
A=[[1,1],[9,6],[12,16]]
b=[200,1566,2800]
x0_bounds=(0,None)
x_bounds=(0,None)
res = linprog(c, A_ub=A, b_ub=b, bounds=(x0_bounds, x_bounds))
print(res)
Output:
Remember the Trickest point to solve this problem is
Min(f(x))= -(Max(f(x))) ⇔ Max(f(x))= -(Min(f(x)))
Therefore we have our Max Profit = 66100 with A-type fibers=122 and B-type fibers=78
fun: -66100.0
message: 'Optimization terminated successfully.'
nit: 2
slack: array([ 0., 0., 88.])
status: 0
success: True
x: array([ 122., 78.])
Visualization:
image
Python Code:
x= np.linspace(0,200,2000)
y1=200-x
y2=(1566-6*x)/9
y3=(2880-16*x)/12
plt.plot(x, y1, label=r'$y\leq 200-x /pre>)
plt.plot(x, y2, label=r'$9y\leq1566- 6x/pre>)
plt.plot(x, y3, label=r'$12y\leq2880 - 16x/pre>)
plt.xlim((0, 200))
plt.ylim((0, 200))
plt.xlabel(r'$x/pre>)
plt.ylabel(r'$y/pre>)
# Fill feasible region
y5 = np.minimum( y1, y2 )
y6 = np.minimum(y2, y3)
y7= np.minimum(y5,y6)
plt.fill_between(x,y7, color='grey', alpha=0.5)
plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)
plt.show()
Reference:
Bellur Srikar, HU ANLY 705, Linear Optimization Lecture notes
http://www.vision.ime.usp.br/~igor/articles/optimization-linprog.html
http://benalexkeen.com/linear-programming-with-python-and-pulp-part-1/
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.linprog.html
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