which also involve multiple vehicle-vessel choices, now allows us to discuss the
question of the relationship between the haulage distance and the optimum vehicle-
vessel size. In order to do this we simply need to calculate the optimum shipment size
under this more general transport rate specification, given that Qi= Qi* under
conditions of vehicle-vessel full-load carrying capacity.
6. Proof of the Relationship between the Optimum Vehicle-vessel Size, the Haulage
Distance and the Structure of Transport Rates
PROPOSITION 1: The theoretical optimum vehicle-vessel size is not always
positively related to the haulage distance, although empirically this is the usual result.
Proof of Proposition 1
If we incorporate space costs and our general description of transport costs into our
initial total logistics costs expression for shipments given in equation (1) we have:
a
v I∩ Idi(b + zɔ) Qi ∩
miSiI ci Qi Qi i si Qi a
TLCi = Q + 2 + 2 + 2 + midi(b + Q )
(8)
which rearranges to:
TLCi=
mi
(Si + a di) Qi [I(Ci + bdi + s,∙)]
Q + 2
+d i
( Ia ɔ
+ mib
I 2 J
(9)
Once again, differentiation and setting to zero gives:
δ( TLCi ) _ mi( Si + a di ) [ I ( Ci + bdi ) + Si ]
δQ∣ ~ ~ Q^ + 2
(10)
which gives an optimised shipment size as:
* _ 2mi( Si + a di )
Qi "V[I(Ci + bdi) + s#]
(11)
In order to determine how the optimised shipment size is related to the haulage
distance we need to take the derivative of (11) with respect to haulage distance thus:
∂Q = - Ib [2mi (Si + adi)]1/2 +a21/2m1/2
ddi 2[I( Cj + bdi ) + Si ]3 2[2 mi( Si + adi )] [I ( Ci+ bdi ) + Si ]
(12)
What we see is that the behaviour of the optimised shipment size, and consequently
the optimum vehicle-vessel size, with respect to haulage distance, depends on the
relationship between the vehicle-vessel movement costs and the vehicle-vessel
carrying capacity, as defined by the values ofa and b.
13
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