di . Thus, assuming di is held constant, as mi increases, the optimum delivery
frequency mi∕Q*i will increase in proportion to∙x∕m^ and the total batch distance
travelled per unit of mi will fall with respect to 1/ ∙x∕m^. Therefore, the fall in ti will
be with respect to 1/ -y∣mi as mi increases. This is the nature of quantity discounts. A
similar line of reasoning can be used to analyse the impact on ti of a change in the
distance di between the input source and the location of the market. Assuming a
fixed volume mi of material is to be moved through a distance di per time period,
we can see that as di increases, the optimum batch size will increase with respect to
dd^i, the delivery frequency will fall with respect to 1 / y[d^i, and consequently, so will
the value of the parameter ti . Thus the distance increase is somewhat offset by a rise
in batch size. This is the nature of economies of distance. This fall in ti will continue
until the value of di is large enough to mean that all goods are shipped in a single
batch per time period5 beyond which there will be no further fall. The point here, is
that even for a single vehicle which exhibits linear movement costs with respect to
distance, as long as the carrying capacity of the vessel is not exceeded, the optimum
shipment size means that the transport cost per ton-mile falls with respect to the
square root of both the haulage distance. Therefore, under these conditions, for a given
ton of shipment, the ton-mile transport cost and the transport cost per ton will both
rise in proportion to the square root of the haulage distance, as has often been
observed (Isard 1951; Tyler and Kitson 1987; Bayliss and Edwards 1970). This
observation provides a partial reconciliation between the paradox suggested by Fig. 1
and 2, for the case of a single vehicle-vessel.
However, in order for this observation to become a general principle we must
be able to show that this ‘square root law’ (Baumol and Vinod 1970) holds for
transport rates when we also any incorporate port-terminal handling costs which are
related to the size of a shipment, and also the question of multiple vehicle-vessel sizes.
Variable port-terminal handling costs are incorporated into our model in a
straightforward way in section 6 onwards6. The next section explains how the insights
from this section into the relationship between the optimised shipment size and the
structure of vehicle-vessel transport costs can also be generalised from the case of a
single vehicle-vessel to the case of multiple vehicle-vessel choices.
5. Generalised Transport Rates with Multiple Vehicle-Vessel Choices
So far we have ignored is the question of the use of multiple vessel or vehicle types on
the behaviour of transport rates within the shipment size optimisation problem.
Indeed, the assumption that a transport or logistics planner is constrained to use a
At this point, the total costs of transportation become greater than the total inventory holding costs.
This is a corner solution. As di is increased beyond this critical value, the frequency of delivery cannot
fall any further and so the value of ti cannot fall any further and will remain constant beyond this
point. Thus, where all goods mi are shipped in a single batch delivery, ti will remain both constant
and a minimum as di changes.
6 See Appendix A.
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