Defining a constraint on time when time is an index to other variables

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Charitha Buddhika Heendeniya

July 12, 2022 14:35 Edited

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Dear Community,

I have defined the following variables.

# Binary indicator variable: 1 if bus b is leaving node i at time t, else 0
n0 = model.addVars(n_cars, n_nodes, time_steps, vtype=GRB.BINARY, name='n0')

# Binary indicator variable: 1 if bus b is arriving at node i at time t, else 0
n1 = model.addVars(n_cars, n_nodes, time_steps, vtype=GRB.BINARY, name='n1')

# Binary indicator variable: 1 if bus i is at node k at time t at node k, else 0
x = model.addVars(n_cars, n_nodes, time_steps, vtype=GRB.BINARY, name='x')

# Binary variable: 1 if bus i moves from node j to k at time t.
z = model.addVars(n_cars, n_nodes, time_steps, vtype=GRB.BINARY, name='z')

# Integer variable: Waiting time (incl. charging time) of bus b at node k, after arriving at the node at time t.
theta = model.addVars(n_cars, n_nodes, time_steps, lb=0, name="theta")

Now I want to write a constraint that says: "The departure time of any car from any node is equal to the arrival time plus the waiting time at that node". 
But so far I was not able to find a way to write this, because time steps are indices to the variables I have defined. 
I also cannot define a new variable for the time, e.g., Continuous variable s(n_cars, n_nodes) is the arrival time of a car at a node, or something similar.
This is because I do not know in advance how many times a vehicle might want to pass through or wait at a particular node. 

Do you have any suggestions for me?
Thank you.

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• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  November 16, 2025 23:33

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  Jaromił Najman

  • Gurobi Staff

  August 09, 2022 09:49

  But so far I was not able to find a way to write this, because time steps are indices to the variables I have defined. 

  The post use a decision variable as an index discusses one possibility of defining such variables.

  I also cannot define a new variable for the time, e.g., Continuous variable s(n_cars, n_nodes) is the arrival time of a car at a node, or something similar.
  This is because I do not know in advance how many times a vehicle might want to pass through or wait at a particular node. 

  I am not sure I understand the issue here. You could define a continuous variable s(n_cars, n_nodes) with some fixed upper and lower bounds defining the min/max waiting time and then let the solver decide what the optimal solution value is. Could you elaborate a bit more on this?

  Best regards, 
  Jaromił

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