This proof implies that as the total quantity per time period of freight to be shipped by
an individual haulier over any given distance increases, the average freight rate falls
with respect to the square root of the individual firm’s market size. Given that the
optimised shipment size is itself a square root function of the total shipment quantity
per time period, this also implies that the cost per ton freight rate charged to individual
customers will generally fall with respect to the square root of the individual
customer’s shipment size. However, there are many exceptions to this principle
dependent on questions of service quality and scheduled freight consolidation
practices, as discussed in Appendix B.
QED
Equations (14), (15), (16) and (17) indicate that the reason why observed transport
costs behave in general as they do with respect to haulage distance and haulage
quantity (Allen 1977; de Borger and Nonneman 1981) is due to the relationship
between the optimised shipment and vehicle-vessel size, and the haulage distance and
haulage quantity, rather than simply the static relationship between the movement
costs, the port-terminal handling costs, and individual vehicle-vessel sizes. Moreover,
this general relationship holds whether we have only a single vehicle-vessel or
multiple vehicle-vessels to choose from, even allowing for the fact that transport rates
will experience discontinuities and integer problems associated with limited vehicle-
vessel choices.
7. Conclusions
Existing theoretical attempts at explaining the relationship between the
optimum vehicle-vessel size, the haulage distance and haulage quantity have always
foundered on the problem of how to incorporate the observed structure of transport
costs into such analyses. This has meant that these questions have been dealt with in
rather a heuristic manner, and when empirical observations are included in such
models the results have been shown to be at best inconclusive. This paper has shown
that it is possible to provide a consistent theoretical explanation of the previously
paradoxical relationship between the optimum vehicle-vessel size, haulage distance,
and the observed structure of transport costs, by combining standard inventory
optimisation techniques with a model incorporating the scale relationships between
vehicle-vessel shipment carrying capacities and vehicle-vessel movement costs. Under
these assumptions, we see that the relationship between the optimum vehicle-vessel
size and the haulage distance depends on the static relationship between vehicle-vessel
movement costs and carrying capacities. At the same time, we see that the observed
structure of transport costs with respect to haulage distance and quantity for each ton
or ton-mile of cargo carried, is itself endogenously determined by the optimum
vehicle-vessel size calculation, and not vice-versa, as has been described in some
previous papers. By calculating the optimum shipment size on the basis of the
movement costs of individual or multiple types of vehicles, transport costs per ton
shipped will always be directly related to the square root of the haulage distance, and
inversely related to the haulage quantity per time period, because of the impact of the
haulage distance and haulage weight on the number of shipments made per time
period for any given geographical distance, i.e. on the optimised shipment-miles
distance, rather than simply on the geographical distance, Moreover, this general
square root observation will hold even if we allow for the indivisibilities associated
18