Home' MHD Supply Chain Solutions : MHD May-Jun 2017 Contents A REVISED MATHEMATICAL DERIVATION OF EOQ
likely reason behind the common mistake
of basing carrying and receiving costs on
total annual costs rather than variable
costs is that the Basic EOQ equation is typically
documented as being based on the total annual
cost of an item.
The traditional EOQ derivation
The following equation is a common textbook
derivation of an item’s total annual inventory cost,
which is subsequently used to derive the common
form of the basic EOQ equation: See equation 5.
r = receiving cost per order
c = annual carrying cost per unit
P = purchase unit price (or production cost
Q = order quantity
D = annual demand quantity
From Equation 5, the EOQ equation is found
by calculating the derivative with respect to
order quantity (Q) and setting the result equal to
0. Doing so yields: See Equation 6.
Solving this equation for Q, the traditional
‘economic order quantity’, gives: See Equation 7.
However, this logic is fundamentally flawed.
Equation 5 suggests that total costs (including
fixed costs) vary based on order quantities, but
they don’t! As this article argues, even if no
items are held (or, more realistically, you have
many out-of-stocks), warehouses, DCs and
stores still have many fixed costs attributed to
them which don’t change.
The revised EOQ derivation
Instead the derivation of EOQ should start from
a determination of total supply chain cost (TC)
across all items, which separates fixed and
varying components as follows: See Equation 8.
TFC = Total fixed costs and
includes the majority of an
organisation’s site, equipment and
TVC = Total variable costs and
includes costs directly related to
all individual stock items; such as
inventory purchase, receiving and
carrying costs. Some of these costs
include a small percentage of site,
equipment and employment costs
which vary with increasing stock quantities on
hand. Equation 8 can also be expressed as
follows: See equation 9.
Where SKUL (Stock Keeping Unit by Location)
i to SKUL n is the aggregate of all individual
SKUL variable costs. From this equation, the
unique supply chain costs attributable to a
particular SKUL awre: See Equation 10.
Where FC* is the relative portion of Fixed
Costs one might allocate to a particular SKUL.
As this portion of total cost is a constant, its
derivation and weighted allocation to each SKUL
is unimportant for the purposes of calculating
EOQ, as you will see shortly (Equation 12). The
right hand side of Equation 10 can then be
broken down as follows: See Equation 11.
Notice that all subcomponents of VCi are
exactly the same as Equation 5 but using only
variable cost components of receiving and
carrying costs (as denoted by the superscript
‘var). As fixed inventory costs are constant,
taking the derivative of Equation 11 and setting
the result to zero yields: See Equation 12.
Equation 12 can then be solved for an
individual SKUL giving Q, the Revised
‘Economic Order Quantity’, as: See Equation 13.
Equation 13 is the premise for this whole
article. Though its differences to Equation 5
appear subtle, they are extremely important in
avoiding incorrect calculations of receiving and
It is worth noting that both the traditional
EOQ formula (Equation 5) and this revised
EOQ formula (Equation 13) are based on the
• Demand is constant over the assessed period
(typically one year). To reduce the impact of
this limiting assumption, assessment periods
can be reduced to lengths of time over which
demand is relatively constant.
• New order quantities (Q) are delivered in
full when inventories reach zero (a min max
replenishment strategy). More complex
derivations of EOQ can be used to reduce
the impact of this assumption, including
implementation alongside a more advanced
time phased replenishment planning (TPRP)
strategy; however, that is outside the scope
of this article.
• Receiving costs are fixed for each order
placed, irrespective of order quantity.
Equation 13 can also be represented
graphically as per the below figure. This figure
(see Figure 1. above) is very similar to traditional
graphical representations of EOQ, but note the
use of variable cost components only.
WHY SHOULD RECEIVING AND CARRYING
COST BE BASED ON VARIABLE COSTS ONLY?
Figure 1. Revised EOQ.
MHD SUPPLY CHAIN SOLUTIONS — MAY / JUNE 2017
SUPPLY CHAIN 39
Links Archive MHD Mar-Apl 2017 MHD July-Aug 2017 Navigation Previous Page Next Page