
The simple EOQ model
attempts to balance holding
and ordering costs Relevant Costs in EOQ
Where:
P
= The ordering cost (dollars per order)
D
= Annual demand or usage of the product (number
of units)
C = Annual inventory carrying
cost (as a percentage of product cost or value)
V = Average cost or value of one unit of
inventory
An example
D = 4,800 Annual
Demand
P = $40.00 Cost to Place an Order
V = $62.50 Value of one unit at Cost
C
= 40% Annual Carrying Cost as a Percentage
Assumptions of the
Simple EOQ Model

A continuous, constant, and known
rate of demand.

A constant and known replenishment
cycle or lead time.

A constant purchase price that is
independent of the order quantity or
time.

A constant transportation cost that
is independent of the order quantity
or time.

The satisfaction of all demand (no
stockouts are permitted).

No inventory in transit.

Only one item in inventory, or at
least no interaction among items.

An infinite planning horizon.

No limit on capital availability.
Dimensions of inventory
models

Deterministic versus Stochastic

Indefinite versus finite planning
horizon

Independent versus dependent demand

Single versus multiple:
+ ”Preserves” the MPS
quantities
+ Suitable for JIT
manufacturing
+ Generates smooth
requirements for material and capacity
– No economic
considerations
– Lumps together
requirements to large orders
– Amplifies lumpiness
through the BOM
– Fluctuating material
and capacity requirements
+ Economic
considerations considering discrete requirements
 Estimation of cost
parameters
– Covering many periods
net requirements tends to create amplified
variability of demand for material and capacity

Physical safety quantity

Used when quantity, demand or
consumption is unreliable

Net req. generated when safety stock
is used (rec. by APICS)

Safety in order quantities

Used when yield is stochastic (e.g.
scrap)

Mainly used in master production
scheduling

Specified fill rate (demand filled from
stock) or Specified service level (probability a stockout will not
occur)

Maximize $ demand filled from a given
investment.

Set SS based on specified number of
Sigmas (Std.Dev., MAD, etc.)

Set SS based on specified time supply.

Minimize shortage occurrences for a
given investment (# of orders with a problem.)

Minimize transaction shortages for a
given investment (# of problems in orders.)

Demand during lead
time is larger than order size.

If ordered only when
replenishment comes, inventory is depleted.

Pink line when
backorders, black when demand is lost.
Examples: KanBan
inventory in PSG or long leadtime processes, like
replenishments from China.
When leadtime
is 5 days, and expected demand is 6 pcs/day, we
order up to 60 + 5*6 = 90. As balance is 48, we
order 42 pcs.

Q is clear but S
somewhat less clear. If we review inventory
balance continuously, then when reorder point is
reached and quantity Q ordered, it will lead to
an expected opening inventory of S

However, if review
is periodic, then the inventory can be more or
less below R, so S would be S = Q +
(Rinventory).

Policy Q,T is
interesting. If demand during review T > Q, this
policy does not really work. Unless we decide
that it is still Q that is ordered but we can
order or more times Q (n*Q).

In practice, Q could
be some physical logistics limit, like full
truckload or a full pallet.

In practice, S could
be some periodic system, like shipping schedule
or production cycle.

Probability of not stocking out during lead time:
Safety stock = Z*δd
Where
– Z
=
value from the standard normal
distribution
– δd
=
standard deviation of demand during
replenishment lead time
Reorder Point = Z*δd + expected demand during
lead time
Examples
– Z(1,645) = 95%
– Z(1,960) = 97,5 %
– Z(3,090) = 99,9 %
Minor
Problems

It
calculates the probability of a stockout during
replenishment lead time, not customer service level
measured as fillrate. They are not the same thing.

What if
delivery time is not certain but a variable, too?

The formula
applies only for normally distributed demand, not other
demand distributions.

How to
incorporate demand forecasts?

Some
Extensions: Variable lead time
If the LT
distribution is binomial, then the joint distribution can be
created manually.
joint δ = √(t*
δ_{d} ^{2}+ d^{2}* δ_{t}^{2}
Where
t
= replenishment lead time
δt
= replenishment lead time variance
d
= demand during average replensihment lead time
δd
= demand variance during average replensihment
lead time

Other Distributions

Normal
distribution may sometimes be a reasonable assumption
due to central limit theorem.

Normal
distribution is symmetric.

Physical
inventory cannot be negative, so standard deviation must
be clearly smaller than average (δd 2< 0,5*X), which
still gives 2,275% negative values for demand.

Skewed
distributions are likewise found in practice. Then
incorrect normality assumption underestimates safety
stock requirements.

There are
statistical tests for testing distributions.

However, it
can prove difficult to find analytical formulas for
other distributions.

One can always
use empirical data and simulate its distribution and
base decision on that distribution.

Consolidation benefits
Var (X + Y) = Var X
+ Var Y

Now that we are talking
about independent inventory management, this theorem
gives a handy way to estimate consolidation benefits for
safety stock.

Likewise, it enables data
conversions: if we have weekly standard deviation and
lead time is two weeks, the theorem above can be used to
calculate twoweek standard deviation.

Probability of
a stockout ≠ customer fill rate

The stockout
magnitude must be weighted with its probability.

In continuous
distributions, this is achieved by a loss function (Vollman
calls it service function).

It is
tabulated in Vollman and approximation formulas for
spreadsheets do exist.
SL = 1 – δd*E(Z) /Q
E(Z) = (1 – SL)*Q/ δd
where
Q = order quantity
SL = Fillrate (service level)
δd = standard deviation of demand (under repl. LT)
E(Z) = partial expectation of distribution (loss
function)

If we use
the values in previous examples, (Q=60, δd = 3*√5) we
get for 95% fill rate E(Z) = (1 – 0,95)*60/(3* √5)

from E(Z) =
0,45 follows (through tabulated values) that Z ≈  0,05

and safety
stock is –0,05 * (3* √5) =  0,34

Reorder
point is demand during lead time + safety stock
(6*50,34) which would become ROP = 29,66.

Parameter
calculation with forecast

Independent
inventory safety stock is held against uncertainty. If
forecasts are correct, no need for safety stock. But
they aren’t.

Forecast
quality is measured with mean absolute deviation (MAD).

If you take
the absolute values of normally distributed, its turns
out that
E(X) = 0,8* δx

So you can
simply replace your demand variance δd with 1,25*MAD in
the previous formulas

Provided,
of course, that your forecast error is ~ N.

JuhaMatti
Lehtonen, Dr. Tech, .TU22.176 OPERATIONS MANAGEMENT,
Helsinki University of Technology, Industrial
Engineering and Management

TPPE37
Manufacturing Control, Department of Production
Economics (IPE), Linköping Institute of Technology
