  In this paper, we introduce a mixed integer quadratic formulation for the
congested variant of the partial set covering location problem, which involves
determining a subset of facility locations to open and efficiently allocating
customers to these facilities to minimize the combined costs of facility
opening and congestion while ensuring target coverage. To enhance the
resilience of the solution against demand fluctuations, we address the case
under uncertain customer demand using $\Gamma$-robustness. We formulate the
deterministic problem and its robust counterpart as mixed-integer quadratic
problems. We investigate the effect of the protection level in adapted
instances from the literature to provide critical insights into how sensitive
the planning is to the protection level. Moreover, since the size of the robust
counterpart grows with the number of customers, which could be significant in
real-world contexts, we propose the use of Benders decomposition to effectively
reduce the number of variables by projecting out of the master problem all the
variables dependent on the number of customers. We illustrate how to
incorporate our Benders approach within a mixed-integer second-order cone
programming (MISOCP) solver, addressing explicitly all the ingredients that are
instrumental for its success. We discuss single-tree and multi-tree approaches
and introduce a perturbation technique to deal with the degeneracy of the
Benders subproblem efficiently. Our tailored Benders approaches outperform the
perspective reformulation solved using the state-of-the-art MISOCP solver
Gurobi on adapted instances from the literature.
