Independent Demand Inventory Formulas
When the demand for an item is independent of the demand for other items, we can treat
the inventory of each item separately. The questions we want to answer for each item are:
When should an order for more stock be placed?
When the order is placed, how many should be ordered?
There are two commonly used approaches to answering these questions. In one case, the
timing of the orders is fixed, but the amount ordered varies from order to order. In the
other case, the amount ordered is kept constant, but the time between orders varies. The
second of these two approaches is called a Q system, the first is called a P system.
1. Q system (continuous or perpetual review system)
based on a constant order size, Q
2 parameters:
Q - order size
R - the reorder point, when inventory drops to this level, an order for Q units is
placed.
This system is also called an ROP (reorder point) system or a continuous review
system. The continuous review refers to the fact, that the inventory level must be
constantly observed to see if it is time yet to place an order.
2. P system (periodic review system)
based on a constant time between orders, P
2 parameters
P – review period or order interval, time between orders
T - target inventory, when an order is placed, the quantity ordered is the difference
between the amount on hand and the target inventory level. Thus, you are in
effect, bringing the level up to the target amount.
This system is also called the FOI (fixed order interval) system, or the periodic
review system. Periodic review refers to the fact that once every order period
(P) you review the inventory level to see how much is to be ordered.
The picture on top of the next page shows a graph of inventory level over time for both
a Q and a P system. The level drops off slowly as demand occurs. When an order
arrives, the level jumps back up in one step. Notice how the amount the level jumps up
is constant in the top graph, the Q system, where as the time between arrival of the
orders is fixed in the lower graph, the P system.
Formulas for the Q system
1. Determining Q
For a company using a reorder point system, there are two ways to come up with a
value for Q. The first way is to choose a value based on convenience, or history. Thus,
if the supplier likes to ship in quantities of 100 at a time, then order 100 at a time. If a
case contains 24 items, and that is enough, order 1 case. In a simple system where
inventory costs are not critical, this approach may be sufficient.
A more sophisticated approach is to compute an order quantity that will minimize costs.
Let's define 6 variables.
C = total inventory cost per year
D = annual demand, treated as constant over the years
H = holding cost per unit held per year
S = order cost per order (or Setup cost, when items are made instead of purchased)
Q = order quantity
B = buffer stock, that is, safety stock
The total inventory cost per year (not including the cost of the items themselves) is
C = annual holding cost
+ annual ordering cost
 average   $ per 
number of $ per






C = inventory unit to +  orders per  order


 level   hold 
 year 
Q

C =  2 + B


H
+
D
Q
(S)
If you were to draw a graph of this equation….by the way, I recently read in the Wall
Street Journal, that 60% of high school students could not draw a graph of a function. I'm
hoping that very few, if any, of you fall into that 60%. Or if you did fall into that category
when you were in high school, by now, with a couple years of college, you could draw
such a graph. That's not to say that I have any assignments that require you to draw such a
graph, because I don't, but I hope you can understand the one shown below… Anyway,
if you were to graph this function showing the cost as a function of the order size Q, you
would get an interesting result. Because the annual holding cost gets bigger as Q gets
bigger, but the annual ordering costs gets smaller as Q gets bigger, the curve goes up on
each end, but has a dip in the middle. The lowest point on the curve is thus a
mathematical minimum point, or, in a practical sense, the point where the total cost is the
lowest. This means there is an optimum order quantity! Remember, this results from
making several simplifying assumptions and we are not including every single type of
cost the system may incur, so this "optimum" order quantity must be taken with a grain of
salt. (Note, the graph uses TC for total cost, where we are using C for total cost.)
You can develop a formula for this minimum order quantity using calculus. (Well, maybe
you can't, but someone could…) Take the derivative of the C formula with respect to Q,
set it equal to zero, and presto, out jumps the following formula:
Q=
2DS
H
This is often times called the EOQ formula, for economic order quantity. A guy named
Ford Harris first figured this out around 1913.
2. Determining the average time between orders. (The text uses TBO for time between orders, but
this is that same thing they call P in the periodic review system. I’ll use P in both cases.)
The time between orders or the order interval, P, is Q/D or Q/d, depending on whether
you want the answer in years or in some smaller unit like months or weeks.
Little d, is the demand per period. If ppy is the number of periods per year, then d=
D/ppy. For example, suppose demand was 520 units per year. If there was 52 work
weeks in a year, the demand rate per week, d, would be 520/52 = 10.
3. Determining R, the reorder point.
The idea of a reorder point is that the order is placed when the inventory level drops to or
below the value of R. But, you may ask, what is meant by inventory level? It seems like
this would be easy to answer. Walk out to the floor, find where the items are kept and
count how many there are. This could be called the on-hand inventory level. When this
level drops below R, order some more. But this could create a problem. On Monday,
you walk out and count the items. There are only 12 left and the reorder point is 15. So
you place an order. It takes three days for the order to arrive. On Tuesday, you go out
and count the items and there are now 10 left. This is less than R, but do you place
another order? No, because you placed an order yesterday and the items are on their way.
We cannot simply compare R to the on-hand level, we must compare it to a level we will
call the inventory position, IP. IP = OH + SR + BO. That is, the inventory position is the
on-hand amount plus the scheduled receipts (ordered but not yet arrived) minus the backorders, those items already promised to customers because we were out of stock when
they demanded them.
An order is placed when IP drops to or below R
To compute R, you must take into account the fact that there is a time lapse between the
time when an order is placed and the time when the order arrives. This time duration is
called the lead time. During the lead time demand continues to occur, so you must plan
to have some stock on hand to cover this demand while the order is on its way. If
demand occurred very regularly and the lead time was always the same, you could
compute the amount of stock needed as (d)(L) where d is the demand rate per period and
L is the lead time in periods. Thus if lead time was 3 days, and you always sold 5 per
day, you could order when you had 15 left and by the time those last 15 were sold, the
order would be in.
If only life were so simple. On the contrary, problems occur that make the lead time take
longer than usual. Demand does not occur in a steady stream; rather, customers arrive at
random, meaning that the rate of demand varies. The way to work with this reality is to
have some safety stock to handle those cases when demand during lead time is greater
than usual. Thus the formula for R is:
R =( d)(L) + B ,
where B is the safety stock (or buffer).
The amount of safety stock to have is determined by the variability in the lead time, the
variability in the demand rate, and the degree to which the management is willing to risk
running out of stock. Statisticians tell us that a good way to measure variability is with
the standard deviation. Let L be the standard deviation of the demand that occurs during
lead time. This depends on the variability of the demand rate, t, and the variability of
the lead time, LT and the length of the lead time. Lucky for you, the text assumes that the
lead time does not vary, so we will make that same assumption. Thus,
L = t L
To compute the level of safety stock needed, we have to determine what probability
of not running out of stock is satisfactory. We find the probability in a normal table
and find how many standard deviations, z, that implies. The safety stock, B, is simply
z*L. By the way, an alternate formula for R is R = dL + ( z)(L ).
Juggling the values in this formula around, you could create another formula that
gives safety stock if R is already known. For example, R might be set by convenience
or by a decision that does not take into account the mathematical probability of
stockout.
B = R - dL
This can be used if you are considering a system currently in place that already has an
R. You could determine how much safety stock is effectively being held by the
choice of R.
Formulas for the P system (FOI)
1. Determining P
Like Q, this can be set in one of two ways. It can be set by convenience, which is
most common, or it can be based on the EOQ formula, to be an optimal P. Since it is
quite common for items to be ordered from companies that provide regular shipments
(such as a warehouse to a retail grocery store), P is usually set to be the time between
trucks in the regular shipment route. Often this corresponds to a common time unit,
like once a week, or twice a month.
If the P is based on the EOQ formula, the formula is:
1 2DS
P=d
H
The result from this formula will often have to be rounded to an integer, since it
would be very difficult to ship something regularly every 4.2924 days, for example.
2. Determining T
The target inventory amount is an amount chosen to provide enough stock to meet
demand until the next order arrives; it is comparable to R in that way. However, you
must think about this carefully. When an order is placed in a P system, the order
arrives after the lead time occurs, but then that amount must last until the following
order arrives which is one more order interval (P) away. Thus the target inventory
amount must last for P + L time periods. (See the text, page 682+ for more details)
Because the time to order is fixed, we cannot order early if there is a rush on the item.
We must wait until the following order time. This will mean that we have to carry
more safety stock with this kind of system. The formula for T is
T = (d)(P + L) + B
B is again safety stock, but safety stock is not equal to the same thing in this system as
it was in the Q system. The equation for B is z times the standard deviation of the
demand that occurs during the lead time and the order interval, or
B = z t
PL
3. Determining safety stock, B
There are two ways to compute safety stock. If you already have computed the target
amount, T, then use:
B = T - d(P + L)
If you do not know T, then use
B = z t
PL
4. Determining the order quantity.
The quantity to order is found by taking the target value and subtracting the current
inventory level or inventory position, IP.
Q = T - IP
The average order quantity (it will vary from time to time) will be (P)(d)
5. The total cost for a P system
d

 D
d

 ppy 
C =   B H  
 S which also equals   B  H  
S
P

 dP 
P

 P 
6. orders per year =
ppy
P
Most of the material I have described here is echoed in the text with only a couple
differences in symbols are noted above. Feel free to contact me about questions you may
have on the formulas and mathematics.