Supply-Chain Optimizer for a demand shock

Order too often and ordering costs pile up; order too much and holding costs balloon. The Economic Order Quantity, EOQ = √(2DS/H), is the order size that minimizes their sum. At the optimum the ordering and holding costs are exactly equal — the crossing point of the two curves. The foundation of inventory management.

The M/M/1 queue models a single server with random arrivals and service — a checkout, help desk, or router. The utilization ρ = λ/μ decides everything: as it approaches 1, the average wait and queue length explode toward infinity. This nonlinear blow-up is why systems run at, say, 80% and not 99% capacity.

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.

The max-flow problem asks how much can be pushed from a source to a sink through a network of capacity-limited edges. Edmonds-Karp repeatedly finds an augmenting path and saturates it until none remain. By the max-flow min-cut theorem, the answer equals the capacity of the cheapest set of edges that, if cut, disconnects source from sink — the bottleneck.