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The optimum shipment size

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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.

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

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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

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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

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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.

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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.

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Model notations

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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

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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

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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

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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:

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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.

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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.

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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).

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