is only appropriate where the firm only has a small quantity of goods to be delivered
to a particular place per time period. In the usual case where a firm consistently has a
very large volume of materials to be moved per time period to a particular set of
locations, it is more economical for the firm to have control over the logistics
operation, irrespective of whether it is using ‘own account’ or third-party haulage
services, and in practice, this is indeed the policy of most large firms.
Appendix C: Proof that the transport rate parameter t cannot remain fixed even with
constant returns to scale in shipment transportation.
Under the hypothetical situation that we experience constant returns to scale in
transportation, we would still not observe that t is fixed in a logistics-costs model
calculated on the basis of the relationship between vehicle movement costs and
vehicle carrying capacities, over a single or multiple vehicle-vessel types. The reason
is that the assumption that t is fixed means that the relationship between vehicle-
vessel movement costs v and carrying capacities Qc is linear. However, within an
Economic Order Quantity model, the assumption that t is fixed means that in general,
v is a linear function of Q*, rather than Qc. Yet, this can never be the case. It is
possible to see this simply by comparing any two types of vehicle-vessels, whose
movement costs are va and vb, respectively, where vb= kva (for any positive constant
k), and whose carrying capacities are Qa and Qb, where Qb= kQa. The relationship
between these two vehicle-vessels exhibits constant returns to scale when we compare
carrying capacity per shipment with the vehicle-vessel movement costs. Assuming
that the logistics planner has a sufficient variety of vehicle-vessel types and sizes that
we can assume there are no less than full-load shipments at the optimum arrangement,
then we can set Qa = Q*a and Qb = Q*b. However, we see that constant returns to
scale in vehicle-vessel shipment movement costs will not also imply that the
transportation costs for any given haulage distance are fixed, i.e. ta≠ tb, because:
va vb va kva
≠ given that ≠^^=^^= ≠
Qa Qb /2 m ( S + dvɑ) /2 m ( S + dkva )
I Ic + Sa I Ic + Sa
Moreover, even if by chance the per ton-mile transport rate happened to be equal for
two different haulage shipment sizes over a given spatial haulage distance, it is clear
that this result would not also hold over haulage distances in general, and that the
transport rate would continually change. This point is important, because it means that
we cannot write a spatial analytical model with a fixed per ton-mile transport rate t=b
parameter as an assumption, as is often done, when we are discussing the question of
frequency and shipment sizes, unless we have both no terminal costs per shipment and
also no economies of scale in the relationship between vehicle-vessel movement costs
and carrying capacities. Under these conditions all goods will be shipped in a
continuous-flow/JIT manner in which t=b irrespective of the haulage quantity and
haulage distance.
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