Assuming that goods are delivered and consumed at a constant rate and there are no
stock-outs or shortages,2 it is initially possible to represent the total logistics costs per
time-period faced by the firm as (McCann 1993;1996):
Total Logistics Costs TLCi =
mi Si IQi (Ci + tidi)
O + 2
+ tidimi
(1)
Port Inventory
Transport Movement
Costs Holding Costs Costs
whereSi represents any fixed component of port or terminal-handling costs which are
independent of the capacity of the vehicle-vessel, but are incurred every time a
vehicle-vessel berths, I is an inventory holding cost coefficient, usually represented as
the sum of the interest and insurance rates, and Qi is the shipment size of the goods or
people (Findlay 1983). Transport movement costs are the fuel and labour-time costs
incurred in moving the goods or people between ports of call. From this we can
determine the optimum shipment size thus:
δ(TLCi) = - mi Si I(Ci + tidi)
δQi ~ Q + 2
and:
2 mi Si
I ( Ci + tidi )
(2)
(3)
From equation (3) we see that the optimum shipment size is a function of the total
transport costs per ton of cargo carried tidi. The particular relationship between the
optimum shipment size and the haulage distance will depend on exactly how total
transport costs change with haulage distance. In this case, it would initially appear that
as long as total transport costs are a positive function of distance, the optimised
shipment size Qi always falls as haulage distance di increases. Clearly, this result is
counter-intuitive, particularly in the light of the empirical evidence (Kendall 1972;
Jansson and Shneerson 1987). As we will see in the next section, however, the reason
we observe this incongruous result is that there is a fundamental analytical problem
with the this approach. This is because in shipment size optimisation problems,
transport rates ti in reality will also be a function of the shipment bundle size Qi ,
while Qi is a function of ti (McCann and Fingleton 1996). The result is that we have
too many variables and the problem becomes indeterminate (Bacon 1984). This is the
reason why spatial pricing techniques have previously been unable to provide any
analytical solution to the problem discussed in section 2. As Bacon (1984 p.115)
points out “the form of the transport cost function plays a key role once we allow
frequency to become a variable. Models in which transport costs are proportional to
bundle size ... yield completely unsatisfactory results, and must be abandoned for
2 Variable demand and one-off deliveries also imply that a variety of individual vessels or vehicles will
be needed at different times. Variable demand uncertainties are usually accounted for by calculating the
optimum buffer stock to add to the simple optimum shipment quantity (Love 1979) and the existence of
these buffer stocks does not fundamentally alter the behaviour of our simple model. The economic
shipment principle will still be used to determine the general logistics arrangements, while one-off
shipments will be organised individually. This means that although our model is set in a steady-state
context, it conclusions are not fundamentally altered by uncertainty.