1. Handbook of Logistics and Distribution Management Sixth Edition Alan Rushton, Phil Croucher & Peter Baker
Useful formulae

2. Contents (1) Concepts of Logistics and Distribution Service quality
Perfect order fulfilment
Procurement, Inventory and Demand Forecasting
Square root law
Inventory calculation with uncertain demand and lead time
Demand forecasting: moving average
Demand forecasting: weighted moving average
Demand forecasting: exponential smoothing
Warehousing and Storage
Cube per Order Index (COI)
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London.

3. Contents (2) Freight Transport
Weight / measure calculation for sea freight
Weight / measure calculation for air freight
Straight-line method of depreciation
Reducing balance method of depreciation
Vehicle running costs: fuel
Vehicle running costs: tyres
Vehicle running costs: total
Vehicle standing costs
Vehicle overhead costs
Zero-based budget
Vehicle cost calculations
Vehicle utilization calculations
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London.

4. Service quality = perceived performance desired expectations X 100%
From the customer viewpoint, service quality is the match between what the customer expects and what the customer experiences - not necessarily what is actually happening in terms of what the supplier is providing (or thinks they are providing).
The basic formula is thus:
Service quality = perceived performance desired expectations X 100%
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

5. Perfect order fulfilment
A common measure of order fulfilment is ‘on time in full’ (OTIF) which measures orders that are delivered both ‘on time’ and ‘in full’. However, the definition of a ‘perfect order’ may include many other additional factors. The basic formula is:
Perfect order fulfilment = number of perfect orders total number of orders X 100%
Example:
orders received on time actual 95% target 98%
orders received complete actual 98% target 99%
orders received damage-free actual 99% target 99%
orders filled accurately actual 97% target 99%
orders invoiced accurately actual 94% target 98%
The actual customer service measure achieved is (95 × 98 × 99 × 97 × 94 =) 84 percent.
This is not as good as it first looks when considering each measure individually.
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 51.

6. Reduction in safety stock = 1 - 5 10 = 1 - 2.24 3.16 = 0.29 = 29%
Square root law
This is used to estimate the potential savings in safety stock as the number of distribution centres (DCs) serving a specific market is changed. The law states that the total safety stockholding in a distribution system is proportional to the square root of the number of depot locations:
Reduction in safety stock = new number of DCs original number of DCs
Example:
A reduction from 10 to 5 distribution centres may lead to a reduction in safety stock of about 29%, as follows:
Reduction in safety stock = = = = 29%
This is based on certain assumptions such as similar sized depots and common products across the whole market.
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 241.

7. Standard deviation of lead time demand = (LT * DSD2) + (D2 * LTSD2)
Inventory calculation with uncertain demand and lead time (1)
Where there is uncertainty in terms of both future demand and lead times, safety stock and reorder levels can be calculated based on the following formula:
Standard deviation of lead time demand = (LT * DSD2) + (D2 * LTSD2)
D: Demand
DSD: Standard deviation of demand
LT: Lead time (from supplier)
LTSD: Standard deviation of lead time
An example is given on the next two slides.
Note: * = multiply
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

8. Inventory calculation with uncertain demand and lead time (2)
Example:
Demand (D) = 100 units per week
Demand standard deviation (DSD) = 20 units
Lead time (LT) = 8 weeks
Lead time standard deviation (LTSD) = 0.5 weeks
The mean lead time demand is LT * D = 8 * 100 = 800 units, and
the standard deviation of LT demand is:
= (LT * DSD2) + (D2 * LTSD2)
= (8 * 202) + (1002 * 0.52)
= units
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p

9. Inventory calculation with uncertain demand and lead time (3)
Example (continued):
For a 95 per cent service level, 1.64 standard deviations are required (from one-tail normal distribution tables)
Safety stock (SS) = 1.64 * 75.5
= 124 units
Reorder level = (LT * D) + SS =
= 924 units
This is based on the assumption that the distributions of demand and lead time are normal distributions.
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p

10. Ft = At–1+ At–2+ At–3+ …+ At–n n
Demand forecasting: moving average
A simple forecasting technique is the moving average, which takes an average of actual demand for a certain number of previous periods and uses this average as the forecast of demand for the next period.
The formula is as follows:
Ft = At–1+ At–2+ At–3+ …+ At–n n
Where:
Ft = the forecast demand
n = the number of periods to be averaged
At–n = actual demand in the past up to ‘n’ periods
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

11. Ft = w1At–1+w2At–2+w3At–3+ …+wnAt–n n
Demand forecasting: weighted moving average
A slightly more advanced forecasting technique is the weighted moving average, which can take account of the age of the previous demands. Thus, more recent periods can be given a higher weighting, as follows:
Ft = w1At–1+w2At–2+w3At–3+ …+wnAt–n n
Where:
w = a weighting factor
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 272.

12. Demand forecasting: exponential smoothing
This is a more sophisticated version of a weighted moving average. This also gives recent weeks far more weighting in the forecast, but each forecast is in fact a weighted average of all prior observations. The weighting process declines exponentially with the increasing age of the observations, thus both emphasizing the importance of the most recent weeks and taking account of the value of the earlier data. The formula for exponential smoothing is as follows:
Ft = aYt–1 + (1 – a)Ft–1
Where:
Ft = the forecast demand
Yt–1 = previous actual demand
Ft–1 = previous forecast demand
a = an exponential smoothing factor (the smoothing constant)
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 272.

13. COI = cubic metres of pick face space number of picks per day
Cube per Order Index (COI)
This may be used in warehouses to determine slotting i.e. which products to place in which locations in the pick face.
COI = cubic metres of pick face space number of picks per day
Example:
SKU 1: COI = 1 m3 / 100 picks = 0.01
SKU 2: COI = 1 m3 / 20 picks = 0.05
The stock-keeping unit (SKU) with the lowest COI (i.e. SKU 1) would be located in the prime picking location.
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 360

14. Weight / measure calculation for sea freight
For sea freight, 1 metric tonne is considered equal to 1 cubic metre.
The price applies to the higher of the two numbers.
Example:
A shipment weighs 1,500 metric tonnes and has a volume of 7,500 cubic metres.
The freight rate is US $75 per weight or measure.
Therefore, the price will be calculated by taking the higher number of the weight or measure and multiplying it by US $75:
7,500 × US $75 = US $562,500
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 450.

15. Weight / measure calculation for air freight
For air freight, 1 metric tonne is usually considered equal to 6 cubic metres (but check with carrier).
This may also be expressed as 1 cubic metre = 0.167t ( = 167 kilogrammes)
- or as 6,000cm3 = 1 kilogramme.
The price applies to the higher of the actual weight or the volumetric equivalent.
Example:
Cargo weighs 150 kilogrammes and has dimensions of 120cm x 80cm x 50cm ( = 0.48 cubic metres).
The freight rate is US $60 per kilogramme weight or measure.
The “volumetric weight” is 0.48m3 x 167kgs = kgs.
This is less than the actual weight, therefore the cargo will be charged based on its actual weight:
150kgs × US $60 = US $9,000
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 479.

16. Straight-line method of depreciation
This is the simplest method of assessing the annual apportionment of the original purchase cost of a vehicle.
Example:
Purchase price of vehicle
53,000
Less cost of tyres
3,000
50,000
Less anticipated residual value
5,500
44,500
Expected vehicle life = 5 years
Annual depreciation (£44,500 / 5yrs)
8,900
See Figure 30.1 to see this in graphical form.
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

17. Reducing balance method of depreciation
This method assumes that depreciation is greater in the early years of a vehicle’s life and becomes less severe in later years.
Example:
£50,000 to be written down
at 36 per cent per annum.
Initial value
50,000
Year 36%
18,000
Written-down value
32,000
Year 36%
11,520
20,480
Year 36%
7,373
13,107
Year 36%
4,718
8,389
Year 36%
3,020
Residual value
5,369
See Figure 30.2 to see this in graphical form.
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

18. Vehicle running costs: fuel
Vehicle running costs are related to an activity (eg distance travelled) and are therefore generally measured in pence per mile or pence per kilometre.
Example:
Price of diesel = 145 pence per litre
Vehicle’s average number of miles per litre = 2 miles per litre
Cost of fuel, in pence per mile: 145/2 = 72.5 pence per mile
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 549.

19. Vehicle running costs: tyres
Tyres are classified as a running cost because tyre usage is directly linked to the distance the vehicle travels.
Example:
Six-tyred vehicle (eg two wheels at front and two double-wheels at back).
Cost of tyres: £500 each
Estimated tyre life: 40,000 miles each
Total cost of tyres: 6 × £500 = £3,000
Tyre cost per mile: = £3,000 x 100p 40,000 miles = 7.5 pence per mile
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 549.

20. Vehicle running costs: total
All running costs can be totalled to give an overall running cost per mile or per kilometre.
Example:
Pence per mile
Fuel
20.0
Oil and lubricants
0.5
Tyres
4.0
Repairs and maintenance
6.0
Total
30.5
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

21. Vehicle standing costs
A second element of vehicle costing is the standing costs (ie those costs that will be incurred over a period of time and do not vary by distance travelled). They are normally expressed for a period of time but can also be expressed for a specified distance travelled over that period.
Example:
Annual standing cost = £9,000
Number of working days per year eg 52 weeks × 5 days = 260 days/year
Estimated annual mileage = 80,000 miles
Therefore, standing costs = £9, days = £34.62 per day
or = £9,000 x 100p 80,000 miles = pence per mile
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

22. Vehicle overhead costs
A third element of vehicle costing is the overhead costs (eg cost of transport manager and transport office).
Example:
Apportioned vehicle overhead = £1,200
Number of working days per year eg 52 weeks × 5 days = 260 days/year
Estimated annual mileage = 80,000 miles
Therefore, overhead costs = £1, days = £4.62 per day
or = £1,200 x 100p 80,000 miles = pence per mile
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, pp

23. Zero-based budget
This is used for setting budgets without reference to previous years’ budgets. This is useful to measure efficiency and to cost a new operation.
Example:
38-tonne GVW 4 × 2 tractor should achieve say 8.5 miles per gallon.
There are six similar vehicles in this fleet.
Their combined annual mileage is 480,000 miles.
The current cost of fuel is say £6.50 per gallon.
Budget = 480,000 miles/8.5 miles per gallon = 56,471 gallons × £6.50 = £367, per annum
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 556.

24. Vehicle cost calculations
Vehicle costs can be calculated on a per case delivered basis or on a per kilometre basis as follows.
Example:
2 rigid £75 per day
150
pence per kilometre
127
3 articulated £146 per day
438
pence per kilometre
205
Total cost per day
920
Cost per case delivered (£920/9,863 cases)
9.3 pence/case
Cost per kilometre (£920/1,296km)
71.0 pence/km
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 593.

25. Vehicle utilization calculations
There are two main utilization measures.
Time utilization = Actual hours used Available hours Load utilization = Actual cases delivered Vehicle capacity in cases
Example:
Time utilization = 44 hours 13 minutes 55 hours = 80% Load utilization = 9,863 cases 11,200 cases = 88%
Source: Rushton, A, Croucher, P. & Baker, P. (2017) The Handbook of Logistics and Distribution Management, 6th Edition, Kogan Page, London, p. 593.