1. The optimum shipment size

2. Empirical observations The observed cost per ton of a given vehicle or vessel is generally linear with distance. The observed freight rates per ton normally diminish with haulage distance (the economies of distance) and with the haulage quantity m i (economies of scale). The optimum size of a vehicle increases with haulage distance.

3. Cost/t is a linear function of distance Cost/t d “Cost-line” of a given transport mode

4. Rate/t “Freight curve” d Z1Z1 Z2Z2 Z3Z3 The freight curve as an envelope to cost lines of z i transport way of increasing capacity z 1 < z 2 < z 3 Economies of distance Transport rates normally diminish with increasing haulage distance, thereby producing a generally concave relationship between transport costs and haulage distance. Freight rate per ton tapering with distance

5. The economies of scale Transport rates fall with respect to the haulage quantity for any given distance thereby producing a convex relationship between transport rates and the quantity shipped m i. transport rate per ton tapering with the haulage quantity m i Rate/t $ mimi

6. The distribution model The model considers a continuous stream of shipments over a given distance sufficient to maintain a constant average annual inventory level and just to avoid a stock out or shortages. The goods are consumed at a constant rate. The time costs of transport of the sum of the individual shipments, is viewed in terms of their aggregate contribution to total annual inventory holding costs.

7. The optimum shipment size The shipment size problem can be defined as determining the shipment size Q* which minimises the total logistics costs LC. The optimisation problem is subject to the constraint that the calculated optimal value of Q does not exceed the total capacity K of the vehicle.

8. Model notations

9. The handling costs Q i is the shipment size of the goods and m i is the total annual shipments. S i represents a fixed terminal-handling costs which are assumed independent of the capacity of the vehicle and of the size of the shipment Q i. The handling costs per year is m i S i /Q i

10. The transport cost The cost v i of moving a vehicle over d i is the cost of fuel consumed plus the labour-hours involved, including where appropriate the return journey. It is assumed that the cost v i is independent on the shipment size Q i carried over a distance d i, if it does not exceed the capacity of the vehicle. The cost of transporting the total annual shipment quantity m i over a distance d i, with shipment size Q i and v i independent of the shipment size, is m i v i d i /Q i

11. The value of an unit of inventory A shipper ships an unit with a price c i per ton over a distance of d i at a transport cost per ton-mile of t i = v i /Q i The transport costs per ton involved can be represented as t i d i and the delivered price can be represented as c i + d i v i /Q i, which represents the value of an individual unit of inventory at destination. I(c i Q i + d i v i )/2

12. The total logistics cost The goods are delivered and consumed at a constant rate and there are no stock-outs. It is possible to represent the total logistics costs LC per time-period i faced by the firm:

13. The square root law The derivation to find the optimum The optimum shipment size thus The size Q i * increases with respect to the square root of m i, and the square root of d i.

14. The optimum frequency The optimum delivery frequency m i /Q i * will increase in proportion to the square root of m i and will fall with respect to the square root of S i and d i.

15. The economy of distance and scale The Q i * shipment means that the transport cost per ton t i d i depends on: Is less than proportional to the haulage distance d i (economy of distance). Is falling with the square root of 1/m i, the haulage quantity (economy of scale).