related to haulage distance. Yet, as we will see in the next section, these are
analytically two quite different problems, and it is these two problems which need to
be unified.
In order to unify these two approaches analytically it is first necessary for us to
compare how shipment optimisation problems are solved:
(a) with transport costs specified as exogenously given freight rates dependent on the
quantity and distance of the bundle of goods to be shipped, as in Fig.1, and
(b) with transport costs specified as endogenously dependent on vehicle-vessel
movement costs, but independent of the quantity and distance of the bundle of
goods to be shipped as in Fig.2.
To do this we must specify transport costs in terms of an explicitly spatial measure
such as transport cost per ton-mile, as is done in spatial pricing models, in order to
allow the endogenous nature of transport cost generation to be investigated. Then it is
necessary to see how these two analytical approaches can be unified initially in the
case of a single vehicle-vessel. Subsequently we will be able to extend the analysis to
the case of multiple vehicle-vessel choices.
3. Shipment Size Optimisation Models
The question of the determination of the optimum size of a shipment and the
associated question of the optimum frequency of a shipment is a well-rehearsed
problem. Here we compare two types of approaches to such problems which explicitly
include distance as a variable.
(a) Models with transport costs related to the quantity and distance of the bundle
shipped
Models in which transport costs are related to the quantity and distance of the bundle
shipped are the usual type in spatial pricing models. In terms of shipment size
optimisation problems these can be represented by the following approach:
For a shipper which ships a quantity mi per time-period of inputs of f.o.b. source price
ci per unit of mi , over a distance of di at a per ton-mile transport cost ti , the total
transportation costs involved can be represented as
tidimi
and the delivered c.i.f. price can be represented as:
ci+ tidi
which represents the value of an individual unit of inventory if the firm is a bonded
carrier. Alternatively, if the shipper is simply a carrier, then the inventory time costs
contribution of the fuel used are given by
tidi
1 The time cost element of fuel used in transit was recognised by Jansson and Shneerson (1982). Our
model combines both the fuel-time and the inventory time costs within the same inventory framework,
whereas they were treated separately by Jansson and Shneerson (1982) and Garrod and Miklius (1985),
respectively.