Time is money: Economies of scale and costing of time-window, last mile delivery... and a special view on last mile food logistics

As of wikipedia (https://en.wikipedia.org/wiki/Last_mile_(transportation)), last mile delivery or transportation describes "the movement of goods from a transportation hub to a final destination", "last mile delivery is an increasingly studied field as the number of business to consumer (B2C) deliveries grow especially from e-commerce companies in freight transportation" and it is not trivial: "some challenges of last mile delivery include minimizing cost, ensuring transparency, increasing efficiency, and improving infrastructure."

In this article, after a quick intro and recap of the 'travelling salesman problem', I will look at economies of scale and the costing of last mile delivery. Then, I will focus on the costing of last mile delivery for specific time windows, say, when a logistics provider commits to deliver 'within the next 2 hours' or '10-12 tomorrow' and close with a summary on when and how a time commitment might make sense economically and some implications for last mile food logistics.

I will focus on costing and I will not look at some other key challenges and complicating factors of last mile delivery like 'recipient not at home', 'returns', 'too heavy for one driver', 'drop factors', 'condition monitoring and management' etc. and focus on stop costing here. I promise not to mention drone or autonomous vehicle delivery beyond this point... ;-)

Travelling salesman problem/ Recap

There is a specific field of operations research for 'last mile' or 'milk run' logistics, when a delivery vehicle travels from stop to stop; questions around finding an optimal/ shortest route covering all stops and the expected length of that route are called 'vehicle routing problems' (VRPs) or 'travelling salesman problems' (TSPs).

Practically, TSPs are solved

• In 2001, a team of mathematicians solved the TSP for 15k cities in Germany (see the pic above this paragraph) - https://www.math.uwaterloo.ca/tsp/d15sol/index.html using a network of 110 processors (*)
• Given that most TSPs have 4-128 stops, respective calculations can be done with standard computers and algorithms today - see for example the google developer route optimization web page https://developers.google.com/optimization/routing/tsp with relevant code in several programming languages and a guide to have google solve TSPs with the 'Google Directions API' (not sure whether that takes actual travel conditions into account (?))

Also, it is known for 60+ years (and is highly plausible based on scaling heuristics), that the expected length of a travelling salesman path (TS path) scales with the square root of the area A that is covered and the number N of stops, so

expected length of TS path ≈ k Sqrt[ A N ]

for large N with some constant k that depends on the area shape and metric (square land/ Manhattan metric vs 'as the crow flies')... and this is what we will use for our costing discussion below.

For 'real world situations' and 'real street networks', k is often set to 1.2 (**).

For square land/ Manhattan metric, it seems that k is close to 1 and that is the value that I will use below.

Costing of last mile delivery

Using the TSP formula and adding a term for the cost of a stop, we find

'milk run' cost = c1 Sqrt[A N] + c2 N

where c1 is a parameter describing the 'travel' cost of driving from stop to stop (cost per km for vehicle, driver, fuel) and c2 describes the cost per stop (driver activity for a stop); as before, for an area A that is covered and N stops.

A top level EU guesstimate for the cost per stop for B2C delivery is c2 ≈ 0.50 EUR (assuming full cost around 15 EUR per hour and 2 mins per stop, we have c2 ≈ 0.50 EUR). See (*3) for an EU guesstimate of the 'travel cost'.

In most situations (unless very dense areas with many stops next to each other), this cost will be less relevant than the 'travel' cost and for the case of specific (and increasingly narrow) time windows that we will look at below, will decrease further... and we will ignore it.

Economies of scale and costing of time-window last mile delivery

For time-window last mile delivery (and small N), we expect 'travel' cost to be most relevant, or

'milk run' cost ≈ c1 Sqrt[A N]

In this approximation, the path length of a milk run should be proportional to the travel time needed.

Now, let's assume, a 'normal shift' covering a full 'milk run' takes 8 hours. Then, a 2 hour time commitment will require a path that has 1/4 the length (of the 8 hour one) to be feasible. If the same area A is covered, then the number of stops has be reduced to 1/16th. Put differently, replacing one 8-hour milk run by 4/ four 2-hour ones (to enable hitting the time commitment), will only deliver to 1/4 the stops and accordingly, 'milk run' costs per stop will increase by a factor 4.

Rephrasing once more as an economies of scale law, we have "sharpening the time commitment by a factor 2 and committing to a delivery in a time window half the size, will increase 'milk run' cost per stop by a factor of 2 (and reduce the number of stops 'produced' per day by a factor of 2)".

Above we are thinking in time commitments that require a separate 'milk run'. Of course, a 'smart' time commitment or rather delivery time prediction, that is based on a standard 8-hour route and a consistent stop density that allows to predict where the delivery vehicle will be at what time of the tour/ day, e.g., "your package will arrive tomorrow between 10 and 12 (as per expected standard route)" or "your package will arrive in the next 2 hours", costs no extra 'milk run' path or time - it 'just' costs a routing and route planning system as well as operational procedures that allow this type of prediction.

Carefully sizing the delivery area, where a time commitment is made/ available, will manage 'travel time' and cost impact and may be required to realize such a service level, e.g., committing to 'service within 2-hours' in an area that is 1/16th of the 'standard size' will allow a similar productivity and costing as the 'standard' case. Think pizza or other hot (or cold) food services with small/ very small number of stops per run and tightly cut areas.

Finally, there may be 'peak load' effects at play: in times of low utilization, e.g., early afternoon (13-16), there may be very little extra cost for time commitments as delivery vehicles are 'idle' anyway; however, customers will normally expect time commitments to be available for different times of the day including peak times (for B2C evening/ not too late, possibly early morning).

... and then there is uncertainty

Now, there is a little, ugly 'secret' about the TSP formula for the expected length of the TS path; not sure it is a 'secret' but it seems to be less known than the formula for the expected value: the variation of a 'milk run'/ travelling salesman path around the expected value is quite sizable for small N.

Based on some numerical experiments the 90% confidence interval of TS path lengths is > 0.7 Sqrt[ A N ] for square land/ Manhattan metric and N<8 and k is in the range of 1.1-1.2; so, for 1 of 10 stop configurations, the actual 'milk run' is >30% off the expected length (shorter or longer).

This of course, has impact on time commitments, costs and planning effort required; probably that is why for actual operations a very practical approach using actual best routes under traffic conditions is used rather than any 'theoretical' value.

It would be great to understand the variability for larger values of N (>20-30)... possibly there is a similar relationship for the standard deviation (*4).

Closing/ Summary

The cost of time commitments in B2C last mile -all else equal- is 'huge' (factor 4 for a 2 hour time commitment vs a single 8 hour milk run)

• Value will only be created in specific circumstances with a respective price premium and willingness to pay: parcel vs express (vs courier) service level
• Cost can/ needs to be managed with geographic area limitations on where that service level is offered - I believe, this (tight postal code constraints on delivery areas) is why hot food delivery on demand is working despite time windows < 1 hour; frozen food works differently with dedicated delivery vehicles (or dry ice/ special packaging set-up) that keep the frozen conditions and does not require a limitation of geographic area - of course, there is value in delivery time prediction: see next bullet; dry and processed foods don't have this limitation and can run with most non-food or any of the food delivery modes; finally, for perishable food, I perceive the need for a time window approach/ commitment and dedicated network to ensure proper temperature management - correspondingly, there is a need for tight geographic area management and additional costs/ a price premium, which may be an element for the 'varying' uptake in different markets. Interestingly, this drives a strong need for localization/ very local service fulfillment vs strong economies of scale in digital customer interfaces as reflected in what is sometimes described as the US 'food delivery war'/ consolidation of 2019-20 (DoorDash, Grubhub, Uber Eats, Postmates etc).
• In many situations, time predictions may mimic time commitments and create two-fold value at limited additional, mostly systemic 1-time cost: (1) reduction of the 'recipient not at home' challenge and cost and (2) potential perception of a dedicated/ premium service

Would you agree?

Many thanks for your input. Stay safe!

Notes

(*) That 'trip' is about 66,000 km long.

(**) Interestingly enough, the value of k is not 'known'/ has not been calculated exactly for even the most basic standard set-ups, e.g., 'as the crow flies' (aka Euclidean metric) on the 1-square in 2 dimensions (with coordinates 'between' (0,0) and (1,1)). Believe it or not, it is an area of ongoing mathematical research.

(*3) I would guesstimate the 'travel' cost of a stop at 3-4 EUR, e.g., with 100 stops per day, 200 working days, 4 km between stops (total of 400 kms), full cost of driver at EUR 15 per hour again, 8 hour shifts and full cost of vehicle (including maintenance) at 40k EUR a year, fuel at 20 EUR for 100km, we find 80k EUR per year with 20k stops. Subtracting the 10k EUR (20k times 0.50 EUR) for stop costs, results in 70k EUR 'travel' cost annually... or (actually, allocating cost in a way that does not quite follow the non-linear economies of scale) 3.5 EUR per stop. Of course, there will be huge variations depending on stop density (area and number of stops), traffic, factor costs etc.

(*4) Does anybody know of relevant research results regarding that question?