I have a binary variable $X_{ijk}$ which I have defined as the following :

X_ijk = [LpVariable("X{0}{1}{2}".format(i+1, j+1, k+1), cat="Binary")
       for i in range(N1) for j in range(N2) for k in range (N3)]

I am struggling to code the following objective function

$$\min \quad \left[\sum_{k=1}^{k=N_3} C^k \left(\sum_{i=1}^{i=N_1}\sum_{j=1}^{j=N_2}P_{ij}^k \cdot X_{ijk}\right)\right]$$

where we know $C^k$ and $P_{ij}^k$. I have defined them as follows :

C_k = numpy.random.uniform(0,100,size=(N3))
P_ijk = numpy.random.uniform(0,100,size=(N1,N2,N3))

Assuming the following has been defined :

prob = LpProblem('Optimization Problem', LpMinimize)

How should I go about modeling the objective function?

  • 1
    $\begingroup$ Hi @Alphanerd and welcome to or.stackexchange. What have you tried so far that doesn't work? $\endgroup$
    – Sune
    Mar 26, 2022 at 21:38

1 Answer 1


I think it would be more convenient storing all the binary decision variables $X$ in a python dict such that x[i,j,k] directly corresponds to $X_{i,j,k}$. Then, because

$$\sum_{k=1}^{k=N_3} C^k \left(\sum_{i=1}^{i=N_1}\sum_{j=1}^{j=N_2}P_{ij}^k \cdot X_{ijk}\right) = \sum_{k=1}^{k=N_3}\sum_{i=1}^{i=N_1}\sum_{j=1}^{j=N_2} C^k \cdot P_{ij}^k \cdot X_{ijk},$$

writing your objective is straightforward:

import numpy as np
from pulp import *

# your definitions of N1, N2, N3 here
C = np.random.uniform(0,100,size=(N3))
P = np.random.uniform(0,100,size=(N1,N2,N3))

prob = LpProblem('Optimization Problem', LpMinimize)

# add the decision variables
X = {}
for i in range(N1):
    for j in range(N2):
        for k in range(N3):
            X[i,j,k] = LpVariable(f"X[{i},{j},{k}]", cat="Binary")

# set the objective
prob.setObjective(sum(C[k]*P[i,j,k]*X[i,j,k] for i in range(N1) 
                                             for j in range(N2) 
                                             for k in range(N3)))

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