Capacitated Facility Location problem with a dynamic set up cost

Question

I am trying to solve a Capacitated Facility Location Problem (CFLP) with a dynamic setup cost in R.

The problem statement is this:

1. I have the transport cost
2. The fixed operating cost (manual labor and other expenses) is known
3. I know the dropping points with loads and all the details
4. The per square ft. cost of rent of a place is known
5. The size of the Facility will be a function of the load. So the rent will depend on how much load is getting allocated in that place.

Assuming the rent will vary like this:

rent= rent_per_square_ft * load* 0.10

Now, I have accommodated the first 4 conditions in my code. But I am not sure how the number 5 can be accommodated.

My model looks like this in R(if it can be of any help):

#m is the number of potential facility/service center (SC) locations
#n is the number of customer locations

model <- MIPModel() %>%
  # 1 if customer i gets assigned to SC j
  add_variable(x[i, j], i = 1:n, j = 1:m, type = "binary") %>%
  
  # 1 if SC j is built
  add_variable(y[j], j = 1:m, type = "binary") %>%
  
  # Objective function
  set_objective(sum_expr(transportcost(i, j) * x[i, j], i = 1:n, j = 1:m) + 
                  sum_expr(fixedcost[j] * y[j], j = 1:m), "min") %>%
  
  # Every customer needs to be assigned to a SC
  add_constraint(sum_expr(x[i, j], j = 1:m) == 1, i = 1:n) %>% 
  
  # If a customer is assigned to a SC, then the SC must be built
  add_constraint(x[i,j] <= y[j], i = 1:n, j = 1:m) %>%
  
  #The demand of customers shouldn't exceed SC capacities
  add_constraint(sum_expr(demand[i] * x[i, j], i = 1:n) <= capacity[j] * y[j], j = 1:m)

I am looking for any headway. Even any link to a relevant article might help.

Answer 1

Based on the comments, the rent for facility $j\in \lbrace 1,\dots,m \rbrace$ would be $0.1\cdot R \cdot \sum_{i=1}^n d_i x_{ij}$ where $d_i$ is the demand for customer $i$ and $R$ is the rent per square foot. If the rent per square foot varies from facility to facility, just change $R$ to $R_j$. Sum those expressions over $j$ and add the sum to the objective function.