Optimal Vehicle Size, Haulage Length, and the Structure of Transport Costs



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 of
a and b.

13



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