Transportation Logistics & Last-Mile Operations
Daganzo Milk-Run Vehicle Routing Length Formula
Estimate multi-stop fleet tour mileage using Carlos Daganzo’s (1984) continuous approximation L_fleet = 2 * r_stem * m_tours + k_tsp * SQRT(N_stops * Area) and quantify consolidation savings.
Estimate fleet delivery mileage in seconds without solving an NP-hard VRP!
Before solving a giant integer-programming Vehicle Routing Problem (VRP), transportation network designers use Carlos Daganzo’s (1984) Continuous Approximation (built on the Beardwood-Halton-Hammersley TSP theorem) to size fleets, depots, and milk-run territories! Serving N_stops customers scattered over a service zone of Area A (km^2) located at average linehaul stem distance r_stem (km) from the depot using vehicles that can visit at most C_max stops per route requires m_tours = ROUNDUP(N_stops / C_max, 0) routes and Total Fleet Tour Distance L_fleet = 2 * r_stem * m_tours + k_tsp * SQRT(N_stops * A), where k_tsp ≈ 0.75 for Euclidean road networks (or ~0.95 for rectangular Manhattan grids)!
Verify on N_stops = 36 suppliers/stores (B2), Zone_Area A = 64 km^2 (B3), Depot_Stem_Distance r_stem = 30 km (B4), Max_Stops_Per_Truck C_max = 12 stops (B5 -> m_tours = ROUNDUP(36 / 12, 0) = 3 milk-run trucks), and Routing_Factor k_tsp = 0.75 (B6): Unconsolidated Direct Point-to-Point Shipping would drive 2 * 30 * 36 = 2,160.00 km, whereas Daganzo’s Consolidated Milk-Run Fleet drives 2 * 30 * 3 + 0.75 * SQRT(36 * 64) = 180.00 km linehaul stem + 36.00 km intra-zone peddling = 216.00 km total (6.00 km per stop — a 90.00% mileage reduction)! Switching to smaller vans capped at C_max = 6 stops doubles stem trips to 6 routes and raises fleet distance to 396.00 km!
| Milk-Run Fleet Design Metric (N=36, A=64 km^2, r=30 km, C_max=12, k=0.75) | Formula | Verified Output |
|---|---|---|
| Direct Unconsolidated LTL Mileage | 2 * r_stem * N_stops | 2,160.00 km |
| Consolidated Milk-Run Fleet Mileage | 2*r_stem*ROUNDUP(N/C_max,0) + k_tsp*SQRT(N*A) | 216.00 km (180 Stem + 36 Local) |
| Consolidation Distance Reduction | (L_direct - L_fleet) / L_direct | 0.9000 (90.00% Saved) |
| Average Fleet Distance per Stop | L_fleet / N_stops | 6.00 km / stop |
Daganzo (1984) Multi-Vehicle Milk-Run Tour Length Formula
Notice the clean decomposition: vehicle capacity C_max affects ONLY the radial linehaul stem term 2 * r_stem * m_tours, leaving the local intra-zone peddling tour k_tsp * SQRT(N_stops * A) invariant!
=LET(n_st, B2, ar_z, B3, r_sm, B4, c_mx, B5, k_rt, B6, IF(OR(n_st < 1, ar_z <= 0, r_sm < 0, c_mx < 1, k_rt <= 0), "Invalid Milk-Run Input", LET(m_tr, ROUNDUP(n_st / c_mx, 0), ROUND(2 * r_sm * m_tr + k_rt * SQRT(n_st * ar_z), 2))))Consolidation Mileage Savings Ratio & Per-Stop Distance Formula
Validate N_stops >= 1, Area > 0, r_stem >= 0, C_max >= 1, and k_tsp > 0.
=IF(OR(ISTEXT(E3), E2 <= 0), "Invalid Milk-Run Input", ROUND((E2 - E3) / E2, 4))Try the example
Use the original values or change them to test your understanding.