I am solving a stochastic facility location model using Benders decomposition (L-shaped algorithm). In each scenario, I want to allocate demands from origin to a fixed number of closest open facilities. For example, I have about 70 candidate facilities and about 370 demand locations (census blocks). I want to make sure in each scenario demand will be allocated to the closest 30 facilities from each demand location to ensure that people do not have to travel too far. If you worked on any similar problem, Would you share how you construct your constraints?
To impose the distance restriction, use a sparse index set of $(i,j)$ pairs rather than the full Cartesian product $I \times J$.
Also, you might consider omitting constraint $(2)$, which will naturally be satisfied unless the penalty for unmet demand is too small to encourage opening any facilities, and constraint $(5)$, which is logically implied by $(3)$ and $(4)$. It is possible for these unnecessary constraints to tighten the formulation, but it is also likely that their inclusion would just artificially inflate the problem size, making the underlying LP solves slower.