Python Day 6:
Linear Optimization with Paper Company
Last time we did a Pirelli Glass Linear Optimization, it considered a simple problem, because it has only 2 variables.What should we do if there are more variables and with much more complexed constrain systems?
diwou / Pixabay
Problem:
Nancy Grant is the owner of Coal Bank Hollow Recycling company, there is two Recycling Process.
Material
- 70 tons of newspaper
- 50 tons of mixed paper
- 30 tons of white office paper
- 40 tons of cardboard
Production:
- 60 newsprint pulp
- 40 packaging paper pulp
- 50 print stock pulp
Let’s use the linear optimation method to minimize the recycle uses and find the most efficient way for recycling.
Process Flow Chart
Solution
Linear Model
MInmize total Cost
f(x)= 13×15+12×16+11×25+13×26+9×35+10×36+13×45+14×46+5×57+6×58+8×59+6×67+8×68+7×69
General Constraints:
all xij>=0
Raw Material constraints :
- newspaper : x15+x16<=70
- mixed paper x25+x26<=50
- White office paper x35+x36<=30
- cardboard x45+x46<=40
Recycling Processes Constraints:
- Recycle Method 1:
0.9x15+0.8x25+0.95x35+0.75x45−x57−x58−x59>=0
- Recycle Method 2:
0.85x16+0.85x26+0.9x36+0.85x46−x67−x68−x69>=0
Production Paper Constraints:
- newsprint : 0.95x 57 +0.9x 67 >=60
- packaging paper 0.9x 58 +0.95x 68 >=40
- stock paper 0.9x 59 +0.95x 69 >=50
Python Code:
import numpy as np
from scipy.optimize import linprog
from numpy.linalg import solve
# Note we need to multiply "-1" in order to change the ">=" sign
A_ub= np.array([
[1,1,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,1,1,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,1,1,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,1,1,0,0,0,0,0,0],
[-0.95,0,-0.8,0,-0.95,0,-0.75,0,1,1,1,0,0,0],
[0,-0.85,0,-85,0,-0.9,0,-0.85,0,0,0,1,1,1],
[0,0,0,0,0,0,0,0,-0.95,0,0,-0.9,0,0],
[0,0,0,0,0,0,0,0,0,-0.9,0,0,-0.95,0],
[0,0,0,0,0,0,0,0,0,0,-0.9,0,0,-0.95]
])
b_ub=np.array([70,50,30,40,0,0,-60,-40,-0])
c=np.array([13, 12, 11,13, 9, 10, 13, 14, 5, 6, 8, 6, 8,7])
best = linprog(c, A_ub= A_ub, b_ub=b_ub,
bounds=(0, None))
best.fun
best.x
Output:
From the result, we can see minimal cost is 1129.94 Dollars, with 1.89 Mixed paper goes to Recycle process 2 and print 66.67 ton newsprint plus 42.1 ton Packing paper and 52.6 ton Stock Paper