IRP - Subtour elimination

[Danny Meier]

Hi,

I implemented a inventory routing problem with Gurobi. There is still one issue I have to deal with.

My current solution produces subtours. I'm not able at the moment to find the right lazy constraint to eliminate subtours.

I had a look to the TSP example how they deal with subtour elimination. But I'm not able to adapt these my problem.

In the callback I have a decision variable y_ij which contains 0 if the edge i->j is not visited or 1 if its visited. A second decision variable is z_i where the value is 1 if the node i is visited or 0 if not.

May you can help me to implement these?

The constraint from the research paper "A Branch-and-Cut Algorithm for a Vendor Managed Inventory Routing Problem" from Archetti et al. look like the equation below:

M are all nodes. T is the time frame (not interesting in this case). But what is S? All nodes from a specific subtour? Or all nodes from all subtours? (Lets say there are two subtours 4-3-4 and 2-1-5-2). M = 1, ..., 5

And what is meant by "for some k"? which k I have to consider?

Thank you very much.
Kind regards,

Danny Meier