Chapter 3 Economic Order Quantity

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Chapter 3Economic Order Quantity
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Defining the economic order quantity
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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).

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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 vary.
• No shortages are allowed.
• Lead time is zero so a delivery is made as soon
  as the order is placed.

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Other assumptions implicitly in the model

• 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.

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• The most important assumption here is that demand
  is known exactly, is continuous and constant over
  time (Fig. 3.2)
• The assumptions give an idealized pattern for a
  stock level. (Fig. 3.3)

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Variables used in the analysis

• Four costs of inventory
• Unit cost (UC)
• Reorder cost (RC)
• Holding cost (HC)
• Shortage cost (SC)

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Three other variables

• Order quantity (Q)
• Cycle time (T)
• Demand (D)

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Derivation of the economic order quantity

• Three steps
• Find the total cost of one stock cycle.
• Divided this total cost by the cycle length to
  get a cost per unit time.
• Minimize this cost per unit time.

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• Amount entering stock in cycle
• amount leaving stock in cycle
• So
• Q D x T
• Total cost per cycle unit cost reorder cost
  holding cost (component)

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• The optimal time between orders is
• To Qo/D 500/6,000 0.083 years 1 month
• 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

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Summary

• 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.

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Adjusting the economic order quantity

• 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
  sizes.
• Deliveries are made by vehicles with fixed
  capacities.
• It is simply more convenient to round order sizes
  to a convenient number.

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How much costs would rise if we do not use the EOQ

• Example
• 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

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Order for discrete items

• This suggests a procedure for checking whether it
  is better to round up or round down discrete
  order quantities
• Calculate the EOQ, Qo.
• Find the integers Q and Q-1 that surround Qo.
• If Q(Q-1) is less than or equal to Qo2, order
  Q.
• If Q(Q-1) is greater than Qo2, order Q-1.

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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.

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Adjusting the order quantity
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or
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or
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Adding a finite lead time

• Time for order preparation
• Time to get the order to the right place in
  suppliers
• Time at the supplier
• Time to get materials delivered from suppliers
• Time to process the delivery

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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

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• (b) Substituting value LT 2 and D 100,
• gives
• ROL LT D 2 100 200 units
• As soon as the stock level declines to 200
  units, Carl should place an order for 250 units.

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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

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• Stock on hand stock on order LT D 300
  units
• Reorder level lead time demand stock on order
• n T lt LT lt (n1) T
• Reorder level lead time demand stock on order
• ROL LT D n Qo

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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.

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Three-bin system

• The two-bin system can be extended to a three-bin
  system which allows for some uncertainty.
• The third bin holds a reserve that is only used
  in an emergency.
• The normal stock is used from bin A.

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• 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.

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• 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.

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• 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 noticing.

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• 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.