a ISi
b I ci+ si
which is the same condition as for equation (12) to be zero. In other words, there is a
unique case where the optimised shipment size and vehicle-vessel size is invariant
with respect to the haulage distance, and where equation (13) must therefore also be
linear with distance, given that the total shipment-distance travelled must be linearly
related to the geographical distance. Apart from this unique chance case, however,
equation (13) is always concave in distance, irrespective of whether we experience
constant or increasing returns to scale in vessel movement costs relative to carrying
capacity. The reason is that the total shipment distance travelled is a function of the
square root of the geographic distance.
QED
PROPOSITION 3: Freight transport rates per ton are always convex with respect to
the total haulage quantity.
Proof of Proposition 3
In order to observe the cost per ton relationship between total logistics-transportation
costs and haulage quantity we take the first and second derivatives of (13) with
respect to haulage distance mi thus:
∂(TLCi/mi) (S ■ a d ) [ I ( c ■ b d ) ■ s ]
(16)
(17)
∂mi ~ 21/2 m3/2
and:
∂2(TLCi/mi) _ 3(Si+ adi)1 [I(Ci + bdi) + ⅜]2^
∂m2 = 23/2 m5/2
Equations (19) and (20) which are negative and positive respectively, define a cost per
ton relationship as described by Fig.6, in which the cost per ton falls with increasing
shipment quantities per time period, although at a decreasing rate.
Cost/ton
The variation in transport cost per ton with respect to the
haulage quantity per time period.
17