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2

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 $(3)$ ...

3

Following is a possible way of its implementation: First, you define a function that takes an OD matrix, solves the GUROBI model, and returns the optimal locations. FUNCTION OPTIMAL_LOCATION(OD): // This part will prepare the gurobi model and change the parameters related to OD matrix // Solve the model and return locations END FUNCTION ...

3

You can add a constraint that says the number of vulnerable people assigned to a facility is at least a specified fraction of the total number of vulnerable people (where the fraction is set to 1 if you want to ensure all vulnerable people are assigned, or something less than 1 indicating your tolerance for leaving vulnerable people unassigned). If you use a ...

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One way to try is to have lower assignment cost (assuming a minimization objective) to vulnerable people compared to non-vulnerable for a given facility.

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