formulation of shipment size optimisation problems can then be employed to provide
proofs to each of the problems concerning the relationship between the optimal
vehicle-vessel and the haulage distance and weight, and also the to prove that total
transport costs will be a concave function of distance even for a single vehicle-vessel,
or for multiple vehicle-vessels irrespective of whether they exhibit economies of scale.
4. A unified approach to the mathematical description of transport rates within
shipment optimisation problems
In order to arrive at a unified approach to discussing transport costs analytically within
the shipment size optimisation framework it is initially necessary to discuss the
behaviour of individual vehicle or vessel transport costs and to see how these are
related to the frequency of shipments made.
In many spatial models, per ton-mile transportation costs are often held fixed
for analytical simplicity. What this actually means is that the transport cost per ton-
mile is independent of the absolute quantity of goods carried. As we see in McCann
(1993; 1998), in the case of individual batch shipment deliveries, this situation cannot
occur. Goods can be carried by transportation forms such as truck, train, ship, airplane
etc. These truckloads, trainloads, planeloads etc. represent the upper limits of the
potential delivery batch sizes. Once the volume of goods per shipment increases above
these size limits, transport-movement costs will generally rise in a stepwise fashion,
since the fixed overheads will rise. If however, we can assume for the moment that
these shipment loads do not exceed the upper physical limits of the particular mode of
transportation (Bunn 1982), we can for the moment assume that transportation costs
are independent of volume carried (Bacon 1984, 1992, 1993). Using a specific form
and size of transportation vehicle-vessel, such as a particular truck or ship, the cost of
moving that vehicle-vessel over a unit distance di will be the cost of fuel consumed
plus the labour-hours involved, including where appropriate the empty or ballast
return journey. This movement cost per mile has already been defined as vi .3 Now
we can assume for the moment that it makes negligible difference to the value of vi
whether the vehicle-vessel carries one unit of mi over the distance di , or one hundred
units of mi over distance di . In other words, as long as the batch size Qi does not
exceed the physical size of the vehicle-vessel, the total cost of transporting the
shipment Qi over a unit distance is independent of the shipment size. The total
movement haulage cost for each shipment Qi will therefore be given by vidi . What
is evident, is that for a given vehicle-vessel, total transportation costs are a linear
function of distance, as in Fig.2.
Now we can imagine two points a fixed distance apart d i . One point is an
input source and the other a market point, with a vehicle-vessel moving forwards and
backwards between these two points shipping a quantity of goods mi per time period
from the input source to the market. The total distance traveled by this vehicle-vessel
per time period is equal to f × 2di where f is the frequency of shipment, i.e. the
number of deliveries per time period. If the market requires 100 units of mi per day,
3 This would be the equivalent of the parameter g in Bacon (1992, 1993) and k in Findlay (1983).