# Under what conditions is the optimal solution to the warehouse location problem the same for a subset of customers?

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:

1. 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?
2. 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.

Edit:

As indicated by the comment earlier. Perhaps to avoid the ambiguity, I would like to define the objective function to my problem as:

$$\min \sum_{i=1}^{n}\sum_{j=1}^{m}d(i,j)x_{i,j}$$

Where there are $$m$$ possible warehouse locations, $$n$$ customers.

• Just to be sure, you are dealing with a $p$-something location problem, right? When you say that you are "minimizing the distance between the warehouse location and its customers" are you talking about the sum of distances (then it is a $p$-median problem) or the maximum distance from a customer to its warehouse (then it is a $p$-center problem)?
– Sune
Jun 25 '20 at 6:57
• That is a very good question, I was talking about the sum of distances, since that seems to be the most simple one. Jun 25 '20 at 19:53