We can describe this relationship in terms of a regression function:
vi =a+bQi
where a represents the positive intercept, given that the smallest vessel appropriate for
industrial haulage purposes will generally incur non-zero movement costs, and b
represents the positive slope, given that in general larger shipment carrying capacity
vehicle-vessels will imply larger movement costs.
Now from our earlier discussion, if the logistics planner ensures that all
individual shipments take place as full load shipments with respect to the capacity of
the vehicle-vessels being employed across the range of vehicle-vessel choices
available, such that the optimum optimorum is always achieved, we can represent a
generalised transport rate function as:
a+ bQ a+ bQ* a
t = ~Q~ = Q = Q +
where a and b are the estimated intercept and slope parameter values across the range
of vehicle-vessel choices available. This transport rate expression is much broader and
more flexible than our previous expression because it allows for transport rates
generated by shipment size optimisation behaviour within a choice set involving
multiple vehicle-vessel sizes and types.
If we have a range of vehicle-vessel choices which exhibits economies of scale in
individual shipment the gradient b will tend towards being zero with a being a non-
zero intercept. Under these conditions the transport rate parameter t will tend towards
being:
t=
aa
over the range of different vehicle-vessel types as well as for a single vehicle-vessel
type, as had been represented by our earlier expression ti= vi/ Qi . The greater is the
value of a and the smaller is the value of b, the more significant are the static
economies economies of scale in shipment transportation. Alternatively, if our choice
range of vehicle-vessels happens by chance to experience constant returns to scale in
shipment size, in our above expression this would be represented by:
and in Fig.5 this will be represented by a straight line from the origin with a positive
slope of gradient b. However, as we see in the Appendix C, in discrete shipment
operations, the presence of the port-terminal handling cost indivisibilities means that it
is not possible to experience an absolutely fixed value of t=b even if we were to
happen experience constant returns to scale in shipment transportation. The only case
where t=b is where there are no port-terminal costs and where we also happen
experience constant returns to scale in shipment transportation. Under these
circumstances, all goods will be delivered in a continuous-flow/JIT manner as
discussed in section 4, and all per ton-mile transport rates will be fixed and
independent of the haulage distance and quantity.
t=
bQ
Q
bQ*
Q*
=b
This more general approach as to how the structure of transport rates in models of type
(a) are generated by optimised shipment calculations from models of type (b), but
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