1. Stocks and Inventories

Chapter 3
Economic Order Quantity
Defining the economic order
Background to the model
The approach is to build a model of an
idealized inventory system and calculate
the fixed order quantity that minimizes
total costs. This optimal order size is
called the economic order quantity (EOQ).
Assumption for a basic model
• The demand is known exactly, is
continuous and is constant over time.
• All costs are known exactly and do not
• No shortages are allowed.
• Lead time is zero – so a delivery is made
as soon as the order is placed.
Other assumptions implicitly in the
• We can consider a single item in isolation, so we
cannot save money by substituting other items
or grouping several items into a single order.
• Purchase price and reorder costs do not vary
with the quantity ordered.
• A single delivery is made for each order.
• Replenishment is instantaneous, so that all of an
order arrives in stock at the same time and can
be used immediately.
• The most important assumption here is
that demand is known exactly, is
continuous and constant over time (Fig.
• The assumptions give an idealized pattern
for a stock level. (Fig. 3.3)
Variables used in the analysis
Four costs of inventory
Unit cost (UC)
Reorder cost (RC)
Holding cost (HC)
Shortage cost (SC)
Three other variables:
• Order quantity (Q)
• Cycle time (T)
• Demand (D)
Derivation of the economic order
• Three steps:
1. Find the total cost of one stock cycle.
2. Divided this total cost by the cycle length
to get a cost per unit time.
3. Minimize this cost per unit time.
• Amount entering stock in cycle =
amount leaving stock in cycle
Total cost per cycle = unit cost + reorder
cost + holding cost (component)
The optimal time between orders is:
To = Qo/D =500/6,000 = 0.083 years = 1
The associate variable cost is:
VCo = HC × Qo = 6 × 500 = $3,000 a year
This gives a total cost of:
TCo = UC × D + VCo = 30 × 60000 +
3000 = $183,000 a year
• We have built a model of an idealized
inventory system that relates order size to
costs and demand. This shows that large,
infrequent orders have a high holding cost
component, so the total cost is high: small,
frequent orders have a high reorder cost
component, so the total cost is also high. A
compromise finds the optimal order size –
or economic order quantity – that
minimizes inventory costs.
Adjusting the economic order
Moving away from the EOQ
The EOQ suggests fractional value for things
which come in discrete units (e.g. an order for
2.7 lorries)
Suppliers are unwilling to split standard package
Deliveries are made by vehicles with fixed
It is simply more convenient to round order sizes
to a convenient number.
How much costs would rise if we
do not use the EOQ
D = 6000 units a year
Unit cost, UC = £30 a unit
Reorder cost, RC = £125 an order
Holding cost, HC = £7 a unit a year
Order for discrete items
This suggests a procedure for checking whether it
is better to round up or round down discrete
order quantities:
1. Calculate the EOQ, Qo.
2. Find the integers Q’ and Q’-1 that surround Qo.
3. If Q’×(Q’-1) is less than or equal to Qo2, order
4. If Q’×(Q’-1) is greater than Qo2, order Q’-1.
Uncertainty in demand and costs
Error in parameters
• Few organizations know exactly what
demand they have to meet in the future.
• The variable cost is stable around the
EOQ and small errors and approximations
generally make little difference.
Adjusting the order quantity
Qo 
2  RC  D
Qo  HC
RC 
2 D
2  RC  D
Q K
2  RC  D
HC  K
Adding a finite lead time
• Time for order preparation
• Time to get the order to the right place in
• Time at the supplier
• Time to get materials delivered from
• Time to process the delivery
Reorder level
The amount of stock needed to cover the
lead time is also constant at:
Lead time × demand per unit time
Reorder level = lead time demand = lead
time × demand per unit time
ROL = LT × D
(b) Substituting value LT = 2 and D = 100,
ROL = LT × D = 2 × 100 = 200 units
As soon as the stock level declines to 200
units, Carl should place an order for 250
Longer lead times
The cycle length is:
T = Q/D = 250/100 = 2.5 weeks
What happens when the lead time is
increased to 3 weeks ?
ROL = LT × D = 3 × 100 = 300
Stock on hand + stock on order = LT × D =300
Reorder level = lead time demand – stock on order
n × T < LT < (n+1) × T
Reorder level = lead time demand – stock on order
ROL = LT × D – n × Qo
Some practical points
• This chapter has shown when to place an
order – set by the reorder level.
• How much to order – set by the economic
order quantity.
• A two-bin system gives a simple procedure
for controlling stock without computers or
continuous monitoring of stock levels.
Three-bin system
• The two-bin system can be extended to a
three-bin system which allows for some
• The third bin holds a reserve that is only
used in an emergency.
• The normal stock is used from bin A.
• Our calculations assume that the lead time
is known exactly and constant.
• In practice, there can be quite wide
variation, allowing for availability, supplier
reliability, checks on deliveries, transport
conditions, customs clearance, delays in
administration, and so on.
• Some stock levels are not recorded
continuously but are checked periodically,
perhaps at the end of the week.
• Then an unexpectedly large demand might
reduce stocks well below the reorder level
before they are checked.
• Another problem appears with large stocks,
such as chemical tanks, coal tips or raw
materials, where the stock level is only
known approximately. Then the reorder
level might be passed without anyone
• We have assumed that the order size is
independent of the lead time and the
reorder level.
• In practice, people often prefer larger
orders if there are longer lead times, and
they raise the reorder level to add an
element of safety.
• The reorder level can be also influence the
order size, with people typically placing
smaller orders with higher reorder levels.

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