In many facility location contexts, it is important to have more than one facility able to cover demands at a node. Thus, in an inventory system that is subject to frequent stockouts, we might like to...

In many facility location contexts, it is important to have more than one facility able to cover demands at a node. Thus, in an inventory system that is subject to frequent stockouts, we might like to have two warehouses capable of replenishing any store in a timely manner. In emergency services such as fire departments, the need for multiple coverage should be clear.

Suppose we want to find the locations of P facilities to maximize the number of demands that are covered by at least two facilities. Use the following notation to show how this problem can be formulated. Also indicate the indices over which summations apply and the indices over which each constraint set applies (e.g., for all i).

Inputs

I = set of demand nodes

J = set of candidate locations

h_i
= demand at node i
 I

P = number of facilities to locate

Decision Variables

X_j
= number of facilities located at candidate site j
 J. Note that this can be any integer number greater than or equal to 0.

MAXIMIZE Total number of demands covered at least twice

SUBJECT TO: Locate at most P facilities

Integrality

And at least one other constraint set which you should define in both words and notation: