Chapter 3 Inventory
INTRODUCTION
While few of us think of ourselves as managing inventory, in fact, we all do. Think about
how you make sure that there are adequate groceries at home – you assess what and how much
you need, how long it will take you to get it and then proceed to purchase it. Or how about gas in
your car – there is some sort of signal to indicate that you need it, hopefully not a sputtering
engine, and usually there is adequate time to actually obtain it. Which leaves you with the
decision on how much to buy – since most of us have credit cards, we opt on filling up the tank.
As a final example, take the issue of cash. How often and how much do we withdraw from an
ATM? There are limits – how much money you have, how much the bank is willing to dispense
each time, and when we will have an opportunity to access another ATM.
Now that you know that you are actually a practicing inventory manager, we can proceed
to take a brief look at the basics of the mathematics behind formal inventory management.
The basic questions of inventory management are:
1. How should the status of the inventory be determined?
2. When should an order be placed?
3. How large should the order be?
These questions are answered by considering four factors:
A. The importance of the item
B. How frequently the inventory is reviewed
C. The selection of an inventory policy
D. What our cost objective is
Before we launch into definitions, it is important to realize that some things are bought to
be used up, or resold, without any further processing. Most consumer goods are in this category
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at the retail and consumer level. Other items are bought to be processed, such as plastic pellets
for molding parts, or onions to be used in a soup.
We will now start with a few definitions that will be helpful in understanding the ideas
behind the formulas:
Replenishment quantity, usually designated as Q, is the amount you order at a given time.
An example would be the two quarts of milk you buy each Saturday. The optimum economic
replenishment quantity, Q* or EOQ, can be determined for most items as we shall see later.
Lead time, usually designated as L, is the amount of time that has to elapse between the
time that you recognize that something is needed and when you actually can get it.
Order point, OP or s, is the quantity remaining when you decide that it is time to order
more. For example, when the little light on your car’s dashboard comes on, there are about two
gallons of gas left and it is time to look for a gas station.
Order up to level, designated as S, is the maximum quantity that you would want or can
have at any given time. Good examples are the size of your gas tank or the space on the
designated shelf for an item.
Raw material is material that has been purchased but has yet to have work done on it.
Work in process, WIP, is the material on which we have started working, i.e., we have
added value to, but have not yet turned into a finished product.
Finished goods are product that has been completed but has not yet been shipped to a
customer.
Demand, D, is the total quantity required per year. For example, a hospital operating
room performs 2000 operations per year, so would require 2000 anesthesiologists’ appearances,
or a car factory manufactures 20,000 of a particular model per year. Demand is also separated
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Chapter 3 Inventory
into Independent Demand, such as the cars in the previous definition, and Dependent Demand,
such as the 80,000 wheels required for those 20,000 cars.
We further distinguish inventory as Cycle Inventory. It exists because time elapses
between successive orders as items are used up. Think of the gas in your tank between full and
the level at which the light comes on and you replenish it.
Another category is Safety Stock (Buffer inventory), designated by SS – the amount we
keep on hand for unpredictable circumstances. We derive this amount from the variations we
expect from the usual usage of an item. If you ordinarily consume one can of tuna a week, you
would buy one can per week. However, just in case the store is out of tuna in any given week,
you keep an extra can at home at all times.
Finally there is Anticipation Inventory – the stock that we build up for a time when the
demand rate will exceed our capacity to produce at the same rate.
Inventory is both beneficial and a substantial expense. You cannot produce or sell
anything without it – that is the good part. On the other hand when it sits around not doing
anything, it can be quite costly. Many factors go into the cost of carrying inventory – space,
insurance, pilferage, obsolescence, the cost of borrowing money to pay for it, etc. This can be up
to about 25% of the value of the item over a year. This is referred to as the inventory carrying
cost, k.
THE ECONOMIC ORDER MODEL
A very simple and nevertheless powerful and idealized model of inventory is shown in
Figure 3-1a. The angled lines represent a constant rate of usage, while the vertical lines represent
the instant replacement of the quantity Q when you run out. From the geometry of triangles we
can tell that the average inventory is Q/2. In Figure 3-1b you can see that if we halve the time
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Chapter 3 Inventory
between replenishments, the average inventory, and the resultant carrying costs are also halved.
So why not replenish even more frequently? The answer is that there is also a cost associated
with each replenishment – placing the order, receiving it, storing it, paying for it are some of the
contributors to this cost and the total cost increases linearly with the number of replenishments.
Figure 3-1. A simplified model of inventory quantity over time
We are now in a position to derive what is the simplest, most used, and oldest concept in
inventory management – the Economic Order Quantity (EOQ) by minimizing the total cost over
time and using our definitions from earlier in the chapter. We just need to define two more
variables, the cost of the item, C, and A, the cost of ordering a replenishment.
Total Cost = DC +
4
QkC
2
+ DA/Q
Equ.3-1
Chapter 3 Inventory
The total cost consists of the direct cost of the item times the demand, plus the holding
cost of the average inventory plus the cost of ordering. Note that we order D/Q times per year.
When we differentiate this equation, set it equal to zero to obtain the minimum and solve for Q,
we obtain:
2AD
EOQ = √ kC
Equ.3-2
An example will demonstrate the reasons that this equation has and continues to
dominate all discussions of inventory management. Suppose a store such as Costco sells 800
55inch big screen TVs per year. Each of these costs the store $300. Each time they order a group
of TVs, a cost of ordering of $80 is incurred. Also, the inventory holding cost is 20% per year.
The resulting EOQ is:
2(80)(800)
EOQ = √(20%)(300) = 48.2
Equ.3-3
Naturally, we cannot order 48.2 TVs. We can round to 50, which would mean ordering
800/50 = 16 times per year. It also might be more convenient to order every 4 weeks, or
800/(52/4), i.e., approximately 60 TVs at a time. The question is, how will that affect the total
cost? To answer this, it is convenient to plot the two costs (note that the DC term is a constant
and does not figure in our EOQ equation) as a function of quantity. You can see that the total
cost curve is very shallow and as long as the selected quantity is reasonably close to the optimal
quantity, the cost does not vary very much. This flexibility in choosing the quantity is very
important – it allows us to accommodate requirements such as package sizes, minimum orders,
truckloads and similar limitations.
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Chapter 3 Inventory
SAFETY STOCK
Our model thus far has been greatly simplified. It is time to add some refinements. The
first of these is the realization that it is seldom possible to have instant replenishment when the
item runs out. So we have to order before we run out. Or consumption continues while we wait
for the replenishment to arrive. This is the lead time. We place our order when our supply
reaches a certain point – this is the Order point (OP) as shown in Figure 3-3.
Figure 3-2 Total Cost as a function of replenishment quantity
The order point is determined by the lead time and the rate of consumption during that
time:
OP = DL
Equ.3-4
In our example we will assume that the supplier of TVs to Costco, perhaps Visio, quotes
a lead time of 2 weeks. Our order point becomes 800(2/52) = 30.8. We usually would round this
up to 31. How do we know that we have reached the order point? Do you constantly look at the
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Chapter 3 Inventory
dash board of your car to see if it is time to buy gas? Well, sort of – you glance there
periodically. Fortunately, in most business we have computers, point of sale devices, etc. to
monitor our stock level and react when the order point is reached. We will examine the situations
when this is not the case a little later, but first we have to also deal with uncertainties.
Figure 3-3 Order point and lead time
We have assumed a constant rate of usage D. Unfortunately, in reality this varies. If we
assume a normal distribution and estimate the standard deviation, we can use this, coupled with a
desired stock out rate, to determine the level of safety stock required. A stock out occurs when a
customer asks for a product and we do not have any. The stock out rate is the percentage of
ordering cycles that will experience a stock out. Let’s use 5% so that the associated z level for
100% - 5% is 1.65. Past experience has shown the retailer that the annual demand of 800 has a
standard deviation σD = 20. But we are interested in the standard deviation only during lead time.
The variation prior to reaching the order points is irrelevant to determining the safety stock.
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Chapter 3 Inventory
σDL = σD√L
Equ.3-5)
SS =z σDL
Equ.3-6)
In our example SS = 1.65(20) √2/52 = 7. This is the number of TVs we keep as
insurance against running out. However, we can expect to have 5% of the cycles to experience a
stock out. Since we have chosen to order every four weeks, or 13 times a year, our expected
stockout will occur every 20/13 years, about every year and a half due to variations in demand,
even with the protection of safety stock as shown in Figure 3-4. The increase in annual cost is
7(300)(20%) = $420.
Figure 3-4 Safety stock added to our model
While suppliers promise a specific lead time, and frequently deliver within it, sometimes
they do not. So it is useful to assume that the lead time is distributed normally and to know the
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Chapter 3 Inventory
standard deviation of the lead time, σL. We can use this information to increase our safety stock
to also allow for this variation. Suppose that the supplier of the TVs indicates (or we glean from
experience) that the standard deviation of the lead time is one day. Combining the variations (the
bars indicate the average of lead time and demand during lead time):
1
adjusted σ = √σ2dl L̅ + σ2L ̅̅̅
d2 =√(16)2 + 7 ∗ (312 ) = 8
Equ.3-7
and our new safety stock is 1.65(8) = 14.
PERIODIC REVIEW
So much for the case when the inventory level is being tracked for us. But suppose we
can only check periodically, such as a weekly inventory assessment. We call this Period Review
and refer to the elapsed time between reviews as R. This increases the period of uncertainty in
the demand to the review period plus the lead time. This method also usually makes use of the
idea of Order up to Level, S. we introduce one more concept, that of Inventory Position. Instead
of only considering inventory already physically on hand it adds any amount ordered, but not yet
received. See Figure 3-5. The wavy lines here indicate that the constant rate of usage is only an
approximation.
σDL = σD√L + R
Equ.3-8
R = Q/D
Equ.3-9
Demand during total lead time = (R + L) * D
Equ.3-10
SS = z𝜎𝑑𝐿
Equ.3-11
S = SS + (R + L) * D
Equ.3-12
Continuing with our example, R = 60/800 = 0.075 or 4 weeks. σDL = 6.8 and SS = 12 and
finally, S = 12 + (2 + 4)*800/52 = 105. Because the order quantity is usually different every
time, it is difficult to estimate the inventory carrying cost without resorting to simulation. We
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will look briefly at one more model – one that combines the review period with an order point.
The effect of this is that if at the end of the review period, we only order if we are below the
order point. Calculations for this model are only possible with trial and error.
Figure 3-5 Order up to level
This has only been a short introduction to the management of inventory, there are whole
long books written on the subject (Silver). However, it should be enough so that you can
appreciate the role of inventory as we explore planning and scheduling in the following chapters.
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