Notebook 3.2 - Transportation Problem (Explicit) using PuLP¶
We are now going to demonstrate the use of PuLP with a slightly larger problem. For this notebook, we will be considering the explicit transportation model formulation that we discussed in Part 2 of our session.
Since we pip-installed pulp while executing our previous notebook, there is no reason to repeat the process again.
Our first step will be to import PuLP and initialise a model instance. As this is a minimisation, we are again using the LpProblem() function to initialise our model.
from pulp import *
prob = LpProblem('problem',LpMinimize)
On this particular formulation, we are being given a lower-bound value for each decision variable, given the presence of the non-negativity constraints. We will therefore use the lowBound parameter when initialising each variable.
x_s1d1 = LpVariable("x_s1d1", lowBound = 0)
x_s1d2 = LpVariable("x_s1d2", lowBound = 0)
x_s1d3 = LpVariable("x_s1d3", lowBound = 0)
x_s2d1 = LpVariable("x_s2d1", lowBound = 0)
x_s2d2 = LpVariable("x_s2d2", lowBound = 0)
x_s2d3 = LpVariable("x_s2d3", lowBound = 0)
x_s3d1 = LpVariable("x_s3d1", lowBound = 0)
x_s3d2 = LpVariable("x_s3d2", lowBound = 0)
x_s3d3 = LpVariable("x_s3d3", lowBound = 0)
We now begin populating our optimization problem.
Our objective function is:
$6x_{S1D1}+3x_{S1D2}+8x_{S1D3}+7x_{S2D1}+5x_{S2D2}+9x_{S2D3}+4x_{S3D1}+5x_{S3D2}+5x_{S3D3}$
prob += 6 * x_s1d1 + 3 * x_s1d2 + 8 * x_s1d3 + 7 * x_s2d1 + 5 * x_s2d2 + 9 * x_s2d3 + 4 * x_s3d1 + 5 * x_s3d2 + 3 * x_s3d3
We can now add the constraints.
prob += x_s1d1 + x_s1d2 + x_s1d3 == 300
prob += x_s2d1 + x_s2d2 + x_s2d3 == 300
prob += x_s3d1 + x_s3d2 + x_s3d3 == 100
prob += x_s1d1 + x_s2d1 + x_s3d1 == 200
prob += x_s1d2 + x_s2d2 + x_s3d2 == 200
prob += x_s1d3 + x_s2d3 + x_s3d3 == 300
You might not be able to tell by simply reading this notebook, but typing all these equations is incredibly tedious and repetitive.
It is also a very error-prone process - it took me 3 attempts to type the model correctly, as I had made several mistakes in quite a few variable names.
We can now print the model formulation to check that it corresponds to the one that we were trying to implement.
print(prob)
It all appears to be correct, and we can now solve oue problem instace using the solve() command.
status = prob.solve()
print(f'STATUS\n{LpStatus[status]}\n')
As before, we are using the following block of code to print the result of our model.
print('SOLUTION:')
for v in prob.variables():
print(f'\t\t{v.name} = {v.varValue}')
print('\n') # Prints a blank line
print(f'OBJECTIVE VALUE: {prob.objective.value()}')