*Theory*

To illustrate the problem in its simplest terms, consider a two-dimensional route delivery example with n points uniformly distributed over a unit square. Beardwood et al. (1959) showed that the average distance per delivery for an optimal route of *n* uniformly distributed locations on a unit square converges to ß/\(\sqrt{n}\) for some finite ß. In other words, a quadrupling of density, like that brought about by a merger of four delivery companies, would result in a 50% reduction in distance traveled, provided the truck had the capacity to deliver the packages. But with capacity constraints problem becomes more complex.

We illustrate some of this complexity in Fig. 1 by considering different truck capacities delivering to *n = 100* randomly distributed points on a unit square. In the left column, we compute distance for a truck that can carry only one package (*k = 1)*, out from a central hub and back again. This out-and-back benchmark can be used as a proxy for the high costs of individual shopping trips relative to home delivery (Brown & Griffida, 2014; Astashkina et al., 2019; Houde et al., 2021). Analytically, out-and-back total distance travelled can be computed as 𝐷1(𝑛) = 𝛼𝑛, 𝛼 ≈ 0.765, where *n* is the number of deliveries.

In the far-right column, we calculate the optimal route for a truck that can deliver all packages in a single day with no driver breaks. For this unconstrained case (*k=∞*), distance travelled is 90% shorter than the out-and-back benchmark, where 𝐷∞(𝑛) ≈ 𝛽 𝑛; 𝛽 ≈ 0.73. In the second row of the table, we see that these savings are not sensitive to the choice of a Taxicab vs. Euclidean distance metric.

In the middle column, we calculate the intermediate case with nine trucks, each serving a single neighborhood “cluster.” Formally, imagine dividing a square area into (*m x m*) neighborhoods with average truck capacity of *k ≈ n/m**2**=𝛼 m**2**+m D**∞**(n/m**2**) = (𝛼/k) n + D**∞**(k)* \(\sqrt{n/k}\). Using the same delivery locations, we see that the distance travelled is 84% shorter than the out-and-back benchmark.

*Application: Austin, TX*

Whether such savings would extend to more realistic Vehicle Routing Problems, with constraints on truck capacities, driver times, and delivery windows is an open question. We answer it by computing optimal routes for randomly generated delivery locations in Austin Texas MSA, using actual street-level expected travel times for two thousand, five thousand, and ten thousand daily deliveries.

We start by simulating random delivery distributions across the Austin MSA, subject to the constraint that each location be a valid delivery address. Then, we cluster those locations using a geographical partitioning algorithm (Carlsson, 2021) and compute optimal routes within each cluster. K-Means geographic clusters are computed subject to the restriction that a zip code cannot lie in more than one cluster.

The clustering algorithm is illustrated in Table 2 for 2500 daily deliveries. Inputs to the algorithm are a collection of locations, a GIS Distance Correlation Matrix (DCM) that provides the travel time between any two locations in the corpus, and a desired number of clusters, K, chosen to satisfy the constraints on driver time, delivery windows, and truck capacity.

The algorithm begins with seed locations that start the formation of the clusters. These can be randomly placed, evenly spaced, or follow some heuristic if other bits of information are to be considered, such as census data block or zip code centroids. Formally, we divide *n* observations into *k*-mean clusters in which each observation belongs to the cluster with the nearest mean (center). Such *k*-means clustering minimizes within-cluster variance (squared Euclidian distance within a cluster).

Then, for each point in the corpus, the closest cluster is determined without violating restrictions, and that point is placed into that cluster. The minimum point in each cluster, according to the sum of the distances of that point to every other point in the cluster, is computed and updated with each point added. This point becomes the seed location for each respective cluster and the algorithm repeats until no new seed locations are determined or a maximum number of iterations has been reached.

After we compute geographical clusters, we assign vehicles to the various clusters and optimize routes. The map to the right shows the clusters and the street-level routes that result. The bottom panel of Table 2 is a sector with a single van route for a density of 2,500 deliveries in Austin per day. To simplify the analysis, we assume that delivery vans have sufficient capacity to make the requisite deliveries to within each neighborhood.

We show the economies of route consolidation in Table 3. In the top panel of the table, we plot the miles/delivery vs. the number of firms. A single firm delivering 2,500 packages (blue line) across an area the size of Austin has: (i) only one-third the miles/delivery as ten firms delivering 2,500 packages (blue line); and (ii) approximately the same miles/delivery as four independent entities making 10,000 deliveries across the same area (grey line).

In the bottom panel of the table, we plot the total distance vs. the number of deliveries. We show: (i) a quadrupling of the delivery density results in a 47% reduction in the miles per delivery, very close to the predicted 50% savings of Beardwood et al.’s (1959) simple model. This translates into huge cost savings, less pollution, and fewer accidents, injuries, and deaths. For an estimated 5,000 daily deliveries in Austin (1.825M/year), a single coordinated delivery enterprise vs. four independent delivery companies results in 203,000 less delivery miles annually.

Table 4

CONSOLIDATING FOUR ROUTES INTO ONE

| ASSUMPTIONS | AUSTIN | US |

**Deliveries** | | 1.825M | 5B |

**Vehicle Costs** | $0.585/Mile | $127K | $348M |

**Driver Costs** | $49,110/Driver year | $148K | $404M |

**Accidents/Injuries/** **Deaths per mile** | 296/84/1.11 | 0.429/0.122/.002 | 1,178/476/6 |

**CO2 Emissions** | .00032 MT/Mile | 66 | 180,000 |

In Table 4, we suppose that a firm like Amazon has consolidated the equivalent of four routes, e.g., by convincing groups of stores to use Amazon’s fulfillment services instead of independent alternatives. We start by assuming 5,000 daily deliveries for Austin as the average number of Amazon deliveries for a city of Austin’s size and quantify the expected reduction in the number of accident/injury/deaths using national rates (296/84/1.11) per 100M miles driven from the National Highway Traffic Safety Administration (NHTSA, 2021, link). Pollution statistics are computed in Appendix B, where we assume that the average freight truck in the U.S. emits 161.8 grams of CO2 per ton-*mile* (Environmental Defense Fund, 2015, link).

Extrapolating from Austin to the rest of the US yields annual savings of 550M delivery miles, emitting 180,000 fewer metric tons of CO2 emissions, with 1,178 fewer accidents, 476 fewer injuries, and 6 fewer deaths, in addition to savings of $348M and $404M in vehicle and driver costs.

*Economies of Density vs. Loss in Competition*

The big economies of the previous section must be weighed against the loss in competition by asking whether stores (and their customers) would be better off had Amazon not consolidated the four routes. On the one hand, stores are attracted by Amazon’s delivery efficiencies but on the other, they are worried that consolidation might increase the price of delivery services. This hypothetical is designed to illustrate the loss of competition vs. economies of density tradeoff.7

To measure the potential loss in delivery competition, we assume that the four consolidated routes belong to four symmetric firms competing in the delivery market with one another. We measure the loss in competition by assuming that all of Amazon’s 21% share of parcel delivery (Pitney Bowes, 2020, link) came from consolidating the four independent firms, who then compete with UPS, FEDEX, and USPS.

To do the benefit-cost analysis, we use a simple Cournot model of competition, and ask whether the lower costs associated with a reduction in miles driven would outweigh the loss in competition. This is equivalent to asking whether the “compensating marginal cost reduction” (CMCR) necessary to offset the anticompetitive effects (Froeb and Werden, 1996) is smaller than the predicted marginal cost reduction. Table 2 illustrates computes the CMCR as a function of the change in concentration and elasticity of delivery demand.

Table 5

% ∆MC that Restores Pre-Merger Price

| Market Demand Elasticity |

∆HHI | -1 | -2 | -3 |

100 | 7.6 | 3.7 | 2.4 |

500 | 18.8 | 8.6 | 5.6 |

1000 | 28.8 | 12.6 | 8.1 |

2500 | 54.7 | 21.5 | 13.4 |

5000 | 100 | 33.3 | 20 |

Suppose that the absolute value of the market elasticity of demand for last mile delivery services is -1 (first column in table above). For our implied change in concentration of 330, it would take at most an 18% reduction in MC to prevent a price increase. Based on our simulations in Austin, the MC reduction of route consolidation is way above that, at 47%. In other words, the economies of route consolidation would lead to a price decrease, not a price increase.