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 Q_i= Q_i^* 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    ∩

^mi^Si^{I           c}i ^Qi              ^Qi ^{i   s}i ^Qi               ^a

TLCi = Q +  2  ⁺      2      ⁺ 2 ⁺ midi^{(b +} Q )

(8)

(9)

Once again, differentiation and setting to zero gives:

δ( TLCi ) _ mi( S_i + a di ) [ I ( Ci + bdi ) + Si ]

δQ∣ ~ ~   Q^   ⁺     2

which gives an optimised shipment size as:

* _   2mi( S_i + 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 +a2¹/²m^1/2

^dd^{i     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.

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