An integrated inventory and routing problem

AHP, a home-care distributor, supplies ten clients on a recurring 12-week planning cycle. Its own distribution system was too expensive and was missing its service level target, so the question was concrete: redesign the single-location system, or outsource the whole thing to a 3PL? And if in-house, which of the two depots should serve which clients?

The model. This is a periodic inventory and routing problem, formulated as a mixed-integer program in AMPL covering a single truck per period, with subtour elimination, depot-selection, periodicity and client-capacity constraints. Seven model versions went into it as the formulation was tightened.

The answer. Serving all ten clients from DC1 costs €5,599.53 over the cycle, DC2 €5,604.63, essentially identical. Outsourcing comes in at €2,102.69 via DC1 and €2,113.57 via DC2. Outsource, and the depot choice barely matters: the 3PL is around 62% cheaper than running the operation in-house, and the difference is not a rounding error in any direction.

Where it gets interesting. Service level was not something the model was allowed to assume away, so it was verified separately by Monte Carlo simulation over 300 scenarios. The baseline network served clients between 61.7% and 89.3%, with the worst client at 61.7%, badly under target. The levers that actually moved it:

• more pallets per visit took one client from 83% to 99.3% and another to 100%.
• halving forecast variance brought 9 of 10 clients to 95% or better.
• raising client capacity pushed every single client above 97%.

So the real finding was that service level is a function of how often and how much you deliver, not of where you deliver from. A second round of analysis tuned the replenishment frequencies directly to a 95% target, with one client still stuck at 62.3% no matter what, which, as far as I am concerned, is the most useful thing the model said.