In this article we focus on approximation algorithms for facility location problems with subadditive costs. As examples of such problems, we present three facility location problems with stochastic demand and exponential servers, respectively inventory. We present a (1+e,1)-reduction of the facility location problem with subadditive costs to the soft capacitated facility location problem, which implies the existence of a 2(1+e)-approximation algorithm. For a special subclass of subadditive functions, we obtain a 2-approximation algorithm by reduction to the linear cost facility location problem.
Gabor, A. F., & Ommeren, van, J. C. W. (2006). Approximation algorithms for facility location problems with a special class of subadditive cost functions. Theoretical Computer Science, 363(3), 289-300. https://doi.org/10.1016/j.tcs.2006.04.013