Facility Location for a tragedy of the commons

Dijkstra's algorithm finds the shortest path from a source (A, blue) to every other node in a weighted graph, always expanding the closest unvisited node next. The number inside each node is its shortest distance from A; the green edges trace the optimal route to your chosen destination. It powers GPS routing, network protocols, and logistics. Click any node to re-target.

Size a graph for the exportable Python solver below — the diagram stays a fixed six-node example.

Linear programming maximizes a linear objective subject to linear constraints. The constraints carve out a convex feasible region (shaded), and the optimum always sits at a corner. Slide the objective coefficients and watch the optimal vertex jump between corners — the geometric idea behind the simplex algorithm that runs global logistics and finance.