I am wondering whether there exists some general insight for when the optimal solution to the warehouse location problem is still optimal for a given subset of customers. For example:
- Does the optimal solution also hold if I remove the customer that is considered an outlier (i.e. the customer that is the further from the warehouse that it's assigned to) from the set of customers?
- What conditions are placed for the possible warehouse locations that are not selected in order for the optimal solution to remain optimal for the subset?
Please note, I am wondering about the case when the objective function simply minimises the distance between the warehouse location and its customers. There are no cost functions related to opening or closing a warehouse. Only the number of warehouses selected is fixed.
As indicated by the comment earlier. Perhaps to avoid the ambiguity, I would like to define the objective function to my problem as:
Where there are $m$ possible warehouse locations, $n$ customers.