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

and if all units are delivered in one single consignment, i.e. Q_i = mi =100, the total
movement costs per day are vid_i . If each unit of mi is delivered individually, as in a
pure JIT mechanism (McCann and Fingleton 1996), the total distance travelled per
day by the ship is 100× 2di and the total transport costs are 100vid_i .⁴ What we see is
that as the delivery frequency increases for a fixed total quantity of material mi to be
shipped per time period, the total transport cost per unit of mi carried also increases.
In this particular case the ratio has increased from(vidi/mi)to(100vidi/mi)as the
number of deliveries per time period increased from one to one hundred. In the
formulations we used earlier, the parameter ti represents the transport cost per ton-
mile i.e. per unit weight-per unit distance. In other words ti represents the cost of
transporting one unit of mi through one unit of distance di . In the example above, in
the first case where there was only one delivery per day, we saw that ti = vi /100 mi
whereas in the second case where there were 100 deliveries per day, ti = vi /mi . In
other words, for a fixed total volume of material, mi , the per ton-mile transport cost
ti , depends on the size of the individual delivery batch shipment Q_i . In this particular
case where the carrying capacity of an individual vehicle-vessel is not exceeded, we
have the general expression ti = vi /Q_i . The reason for this is that for any fixed
geographical delivery distance a change in the delivery frequency implies that the total
shipment-distance travelled per unit of mi changes. In other words, the relationship
between economic distance and geographical distance depends on the shipment
frequency. Bacon (1984) therefore suggests that these issues should be dealt with
sequentially, with location being the first choice and delivery frequency being the
second choice. This is exactly the approach of the inventory and logistics optimisation
techniques employed in reality.

From the above discussion we can reconcile the two shipment size optimisation
approaches (a) and (b) by substituting the expression ti= vi/ Q_i into our equation (1)
to give:

m_τr m∖√ , ./ I^¹^^cI d¹ divi                                       (∏∖

TLCi = q( Si ⁺ divi )⁺ 2 ⁺ 2                                               (7)

which differentiating and setting gives equations (5) and (6) as before. The only
difference is that (7) is actually more comprehensive than (4) in that it also includes
the capital time costs of fuel and labour usage, as discussed in footnote 1. This means
that for a single type of vehicle-vessel, we can convert a shipment size optimisation
problem of type (a) into one which is of type (b), by substituting the expression
ti= vi/ Q_i^* into any model of type (a).

In the case of a single vehicle-vessel, in order to see how transport cost per
ton-mile rates ti typically behave we can observe how ti= vi/ Q_i^* changes with
respect to mi and di . From equation (6) we see that the optimum delivery batch size
increases with respect to the square root of mi , and with respect to the square root of

⁴ A pipeline carrying a fluid has the same mathematical properties.

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