There is not just one facility location problem - there are rather many special cases. If I were you I would start by simply use Google, and check the models that you can find, such that you have a variety to look at - and perhaps one of them is what you need! :-)
Furthermore: the facility location problem is so easy to solve that you shall not need any heuristic - simply run it through a math programming solver for integer optimisation. It's one of the easiest integer network problem there is.
I agree with Michael Patriksson , your problem might be easy enough to solve with a standard solver. First, you have to define the specific facility location problem that you are trying to solve. Then, you need to do a search to find out about the models that are similar to your problem. Finally, you need to formulate your problem according to one of the models and attempt to solve it with a solver. If that doesn't work, then you might need to develop a specialized algorithm (if you have extended a model) or use one from the papers you identified. The following are some recent reviews of facility location models:
Article Extensive facility location problems on networks: an updated review
Article Facility Location Problems: Models, Techniques, and Applicat...
The simplest problem is called the "uncapacitated facility location problem" (UFLP) or "simple plant location problem" (SPLP). It has been studied in great depth since the 1960s. For a review of early works, see this survey paper:
J. Krarup & P.M. Pruzan (1983) The simple plant location problem: Survey and synthesis. European Journal of Operational Research, 12, 36-81.
For a more recent survey, see:
V. Verter (2011) Uncapacitated and capacitated facility location problems. In: H. Eiselt & V. Marianov (eds) Foundations of Location Analysis. New York: Springer.
There is also a huge literature on approximation algorithms (i.e., heuristics that have a performance guarantee). See for example:
S. Li (2013) A 1.488 approximation algorithm for the uncapacitated facility location problem. Information and Computation, 222, 45-58.
Finally, there are some good exact algorithms, such as:
C. Beltran-Royo, J.P. Vial & A. Alonso-Ayuso (2012) Semi-Lagrangian relaxation applied to the uncapacitated facility location problem. Computational Optimisation & Applications, 51, 387–409.
A.N. Letchford & S.J. Miller (2014) An aggressive reduction scheme for the simple plant location problem. European Journal of Operational Research, 234, 674-682.
M. Fischetti, I. Ljubić & M. Sinnl (2017) Redesigning Benders decomposition for large-scale facility location. Management Science, 63, 2146-2162.