Chapter 12
Inventory Models
Introduction
• Inventory management is one of the most important decisions
faced by many companies.
• They all face two competing pressures.
– The first is the pressure to have enough inventory on
hand.
• The most obvious reason for this is that they do not want to run
out of products that customers demand.
• Another prominent reason, however, is the fixed cost of ordering
or producing, as discussed throughout this chapter.
• If a fixed cost is incurred each time the company orders from its
supplier, or a fixed cost is incurred each time a manufacturer
produces a batch, where this cost does not depend on the order
or batch size, the company has an incentive to place large
orders or produce large batches to minimize its annual fixed
costs
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Introduction continued
– The second pressure related to inventory management
is the pressure to carry as little inventory as possible.
• The most obvious reasons for this are the cost of storing items
and the interest costs involved in tying up money in inventory.
• These two competing pressures are at the heart of
most inventory models.
• The balance is typically not easy to find, so they
need models to determine the best ordering (or
production) policy.
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Introduction continued
• An inventory problem can usually be broken up into
two parts:
1. how much to order on each ordering opportunity and
2. when to order.
• When customer demand is assumed to be known, the
resulting models are called deterministic models.
• A more realistic situation occurs when customer
demand is uncertain. In this case, the decision on
when to place orders becomes more difficult. These
more difficult problems require probabilistic inventory
models.
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Categories of inventory
models
• Researchers have analyzed many inventory models,
both deterministic and probabilistic.
• We discuss only the most basic of these models,
which have been used extensively in real applications.
• We begin by discussing several important issues and
introducing some terminology.
• Keep in mind, however, that the possible number of
real-world situations that require inventory
management is virtually unlimited.
• We list only some of the factors that are common to
these situations.
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Deterministic vs. probabilistic
models
• In deterministic models, all inputs to the problem,
particularly customer demand, are assumed to be
known when the decisions are made.
– In reality, a company must always forecast future
demands with some type of forecasting model.
– The outputs of this forecasting model might include a
mean demand and a standard deviation of demand.
– In deterministic models, however, only the mean is
used, and any information about the uncertainty, such
as the standard deviation, is ignored.
– This makes the resulting models simpler, but usually
less realistic.
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Deterministic vs. probabilistic
models continued
• Probabilistic models use this information about
uncertainty explicitly.
• They are typically more difficult to analyze, but
they tend to produce better decisions, especially
when the level of uncertainty is high.
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External vs. internal demand
• A second factor in inventory modeling is whether
demand for the product is generated externally or
internally.
• External demand (or independent demand)
occurs when the company that sells the product
cannot directly control the extent or the timing of
customer demand.
• In contrast, internal demand (or dependent
demand) occurs in most assembly and
manufacturing processes.
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Ordering vs. production
• A third factor in inventory modeling is whether the
company orders the products from a supplier or
produces them internally.
• If the products are ordered, then there is typically
an order lead time, the time elapsed from when
the order is placed until it arrives.
• In ordering models, there is also usually a fixed
cost (also called a setup or ordering cost) each
time an order is placed, where this cost is
independent of the order quantity.
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Ordering vs. production
continued
• In contrast, if products are produced internally,
there is also a lead time, the time it takes to
produce a batch of items. This time is determined
by a production rate.
• As in ordering models, there can also be a setup
cost each time a batch is produced, where this
cost is independent of the batch size.
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Continuous vs. periodic
review
• A fourth factor in inventory modeling is whether
inventory is reviewed continuously or periodically.
• In continuous review models, the inventory is
monitored continually and orders can be placed at
any time.
• Typically, there is a reorder point - a specific
inventory level—so that when the inventory on
hand reaches this reorder point, an order is placed
immediately.
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Continuous vs. periodic
review continued
• In contrast, in periodic review models, there is some
standard time, such as every Monday morning, when the
inventory is reviewed and ordering decisions are made.
• Except possibly for emergency orders, these are the only
times when orders are placed.
• Continuous review models can certainly be implemented,
given today’s computerized access to inventory levels in
real time, and these models can result in lower annual
costs than periodic review models.
• However, when a company stocks many products
(hundreds or even thousands), it is often more convenient
to use periodic review.
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Single-product vs. multipleproduct models
• A final factor in inventory modeling concerns the
number of products involved.
• Models that consider only a single product are
conceptually and mathematically simpler, so we
initially analyze single-product models.
• However, most companies have many different
products that must be considered simultaneously.
• If the company orders these items from a supplier,
it may be wise to synchronize the orders in some
way to minimize ordering costs.
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Types of costs in inventory
models
• Companies face a number of costs when they manage
inventories. Although the types of costs vary depending on
the company and the situation, the following costs are
typical.
• Ordering (setup) cost is the fixed cost incurred every time
an order is placed or a batch is produced, independent of
the amount ordered or produced.
– This ordering cost includes the cost of paperwork and
billing each time an order is placed and could include
other costs as well, such as paying a truck driver to
deliver the order to the company’s warehouse.
– If the product is produced rather than ordered, this cost
can include the cost to set up equipment.
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Types of costs in inventory
models continued
• The unit purchasing (or production) cost is the
cost for each additional unit purchased or
produced (often referred to as the variable cost).
• The holding (or carrying) cost is the cost that
motivates the company to keep less inventory on
hand.
– This cost generally has two components, the financial
holding cost and the nonfinancial holding cost. The
nonfinancial holding cost is usually the cost of storing
the product.
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Types of costs in inventory
models continued
• It is often important to measure the cost of running
out of inventory. This shortage (or penalty) cost is
a difficult cost to measure.
– At one extreme, there are lost sales models, where any
demands that occur when inventory is zero are lost;
these customers take their business elsewhere.
– At the other extreme, there are complete backlogging
models, where demands that occur when inventory is
zero are satisfied as soon as a new order arrives.
– Both of these models—or any in between, called partial
backlogging models—have negative effects for the
company.
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Types of costs in inventory
models continued
• Finally, there is the selling price of the product and
the resulting revenue to the company.
– In many situations, the revenue is a fixed amount that is
not affected by any ordering decisions.
– This occurs when the selling price remains constant and
the company intends to satisfy all demand eventually.
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Economic order quantity
(EOQ) models
• We first examine a class of models called
economic order quantity (EOQ) models. These
are the most basic of all the inventory planning
models.
• We begin by studying the most basic EOQ model.
Then we examine several interesting variations of
this basic model. All of these models make the
following assumptions:
– A company orders a single product from a supplier and
sells this product to its customers.
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Economic order quantity
(EOQ) models continued
– Orders can be placed at any time (continuous review).
– There is a constant, known demand rate for the product,
usually expressed in units per year (annual demand).
– There is a constant, known lead time for delivery of the
product from the supplier.
– There is a fixed ordering cost each time the product is
ordered, independent of the size of the order.
– The price the company charges for the product is fixed.
– The annual holding cost is proportional to the average
amount of inventory on hand.
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The basic EOQ model
• The most basic EOQ model adds the following two
assumptions.
– No stockouts are allowed; that is, the company never
allows itself to run out of inventory.
– The unit cost of purchasing the product from the supplier
is constant. In particular, no quantity discounts are
available.
• These assumptions have important implications.
– Because the demand rate and lead time are assumed to
be known, the company can ensure that it always has
enough on hand to meet demand on time.
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The basic EOQ model
continued
• The main decision is whether to order small
amounts frequently or to order large amounts
infrequently.
• The former results in large fixed costs and small
holding costs (less inventory on hand), whereas
the latter results in the opposite.
• The EOQ analysis balances these two competing
forces.
• The basic EOQ model is demonstrated in Example
12.1
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EOQ model with quantity
discounts
• Example 12.2 illustrates one of many possible
variations of the basic EOQ model.
• In this variation, the company placing the order can
obtain quantity discounts from its supplier.
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EOQ model with shortages
allowed
• A key assumption in the basic EOQ model is that
the company decides, as a matter of policy, not to
allow any shortages.
• Because the demand rate and the lead time are
known, the ordering can be done so that an order
arrives just as the inventory level reaches zero.
• This means that it is possible to prevent shortages
from occurring.
• However, it might be in the company’s best
interests to allow a few shortages if the penalty for
a shortage is not too large.
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EOQ models with shortages
allowed continued
• First, are shortages backlogged or are these
demands lost?
• And what about the penalty cost for a shortage?
• Does the penalty relate only to the number of units
short per year or also to the amount of time the
shortages last?
• Whatever type of shortage cost is assumed, the
practical difficulty is then assessing a specific
dollar value for this cost.
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EOQ models with shortages
allowed continued
• Example 12.3 illustrates a complete backlog model
where the penalty cost is charged per unit short
per amount of time short.
• In this case, the annual penalty cost is a constant p
multiplied by the product of the average number of
units backlogged and the average amount of time
a customer has to wait for a backlogged unit.
• The constant p is the penalty cost charged for
each customer who has to wait one unit of time for
one backlogged item.
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Reducing the setup cost
• There has been a lot of talk in recent years about
striving for zero inventory.
• The argument is that the less inventory a company
carries, the more efficiently it is operating its
business.
• The question is whether this argument can be
justified from an economic point of view, at least in
the context of the EOQ models we have been
discussing.
• One possible answer to this question is to
reconsider whether the setup cost is really fixed.
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Reducing the setup cost
continued
• There are two modeling problems here.
– The first is to choose a reasonable form for the function f(K).
– The second is to find a way to turn a one-time investment
cost, f (K), into an equivalent annual cost, so that the cost of
reducing the setup cost is comparable to the annual operating
costs we have been discussing.
• Putting all of this together, the total annual cost to the
company is f (K)i plus the annual operating cost from any of
the previous models.
• In addition to any previous decision variables, such as Q, K
must be chosen, subject to the constraint K K0.
• Example 12.4 illustrates the procedure.
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Synchronizing orders with
several products
• Until now, all models have considered a single
product.
• If the company orders several products, it could
calculate the EOQ for each product and order
them according to separate schedules.
• However, there might be economies, particularly
reduced setup costs, from synchronizing the
orders so that several products are ordered
simultaneously.
• This should be particularly attractive for products
that come from the same supplier.
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Synchronizing orders with
several products continued
• We develop a model in this section that takes
advantage of synchronization, and we compare it
to the “individual EOQs” policy that uses no
synchronization.
• Although this model can be developed for any
number of products, we keep things relatively
simple by assuming that there are only two
products.
• This model is shown in Example 12.5.
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More than two products
• Virtually the same spreadsheet could be used for more than two
products, provided that we make a simplifying assumption.
• This assumption is that a setup cost reduction is available only
when the company places an order for all of the products
simultaneously.
• Unfortunately, it is probably more realistic to assume that there is
a setup cost reduction when any subset of products is ordered
simultaneously.
• If we allow this possibility, however, and then try to optimize over
all possible synchronizations, the problem becomes difficult to
model in a spreadsheet.
• Therefore, we do not pursue this multiple-product model any
further here.
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Probabilistic inventory models
• In most situations, companies that make ordering and
production decisions face uncertainty about the future.
• Probably the most common and important element of
uncertainty is customer demand, but there can be others.
• The company that faces uncertainty has three basic
options: use best guess, develop a nonsimulation model, or
develop a simulation model.
• The advantage of a simulation model is that it is relatively
easy to develop, regardless of the complexity of the
problem. The disadvantage is that it can be difficult, or at
least time-consuming, to find optimal ordering policies from
a simulation.
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Newsvendor model
• The newsvendor model is one of the simplest
probabilistic inventory models, but it is also a very
important one.
• It occurs whenever a company must place a onetime order for a product and then wait to see the
demand for the product.
• The assumption is that after this demand occurs,
the product is no longer valuable.
• Given the single chance to order, the company
needs to balance the cost of ordering too much
versus the cost of not ordering enough.
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Newsvendor model continued
• To put this problem in a fairly general setting, let
cover and cunder, respectively, be the cost of having
one more unit or one fewer unit on hand than
demand.
• Now let D be the random demand.
– We assume that D has a cumulative probability
distribution F(x), so that for any potential demand x, F(x)
is the probability P(D < x) that D is less than or equal to
x.
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Newsvendor model continued
• The optimal order quantity is given by the following
equation: 𝐹 𝑄 =
𝑐𝑢𝑛𝑑𝑒𝑟
𝑐𝑜𝑣𝑒𝑟 +𝑐𝑢𝑛𝑑𝑒𝑟
• The fraction on the right side of this equation is called the
critical fractile.
– This fraction determines the optimal order quantity
through an examination of the demand distribution.
• For any particular demand distribution, you can appeal to
@RISK, built-in Excel functions, tables in books, or some
other means to find the optimal Q.
• We illustrate the procedure in the following continuation of
the Walton Bookstore calendar example from Chapter 10.
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Example 12.6:
Background information
• Recall that Walton Bookstore buys calendars for
$7.50, sells them at the regular price of $10, and
gets a refund of $2.50 for all calendars that cannot
be sold.
• As in Example 10.3 of Chapter 10, Walton
estimates that demand for the calendar has a
triangular distribution with minimum, most likely,
and maximum values equal to 100, 175, and 300,
respectively.
• How many calendars should Walton order to
maximize expected profit?
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The (R,Q) ordering policy
• The previous subsection analyzed a one-time
ordering decision, which is relevant for a product
such as a newspaper or a fashion item that quickly
goes out of style.
• We now examine an ordering decision for a
product with sales that continue into the indefinite
future.
• Consider this: the probability distribution of
demand in any month, say, is always the same,
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The (R,Q) ordering policy
continued
• As with the deterministic EOQ model, the company
must make two decisions: when to order and how
much to order.
• We assume that it uses a popular type of policy,
called an (R,Q) policy, where R is the reorder
point and Q is the order quantity.
• Under this policy, the company continually
monitors its inventory. When inventory drops to R
or below, the company places an order for Q units.
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The (R,Q) ordering policy
continued
• When a company chooses the reorder point R, it
must take into account the effects of running out of
inventory.
• If the company believes shortages are very
expensive or undesirable, it should choose a
relatively large value of R.
• This leads to a relatively large level of safety
stock, the expected amount of inventory left over the cushion - by the time the next order arrives.
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The (R,Q) ordering policy
continued
• On the other hand, if shortages are not considered
too expensive or undesirable, the company can
afford to use a lower value of R, with a smaller
resulting level of safety stock.
• As in the newsvendor model, we show how to
determine an appropriate trade-off between
leftovers and shortages.
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The (R,Q) ordering policy
continued
• To specify an (R,Q) policy, we must also determine
the appropriate order quantity Q.
• It turns out that the choices of R and Q can be
made almost independently.
– The choice of R depends largely on how shortage costs
(or customer service) are measured, whereas the choice
of Q depends mostly on the same cost factors
considered in the deterministic EOQ models.
• Fortunately, it is possible to develop a Solver
model that determines Q and R simultaneously, as
illustrated in the example 12.7.
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Random lead times
• We have assumed that the lead time for orders is a
known quantity.
• It is not difficult to modify the analysis for the case
where the lead time L is random.
• This is important, because it is not at all
uncommon in real applications for ordering lead
times to be uncertain – suppliers might not be able
to deliver according to a precise schedule.
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Random lead times continued
• When L is random, we need to estimate its mean and
standard deviation, which we denote by L and L. Given
these values the expected demand during lead time
becomes LD =L AD and the standard deviation of demand
during lead time becomes
2
2
 LD   L AD
  AD
 L2
• Completing the calculation, we see that the extra
uncertainty about the lead time adds to the uncertainty
about the demand during lead time.
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Random lead times continued
• Once we use these formulas to obtain LD and LD,
we find the optimal (R,Q) exactly as in the
nonrandom lead time case.
• When the lead time is uncertain, a company needs
to order earlier, which means larger safety stock
and higher inventory holding costs.
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Ordering simulation models
• As problems become more complex, the required mathematical
models become too difficult for most managers to comprehend.
• In fact, mathematical models do not even exist for many realistic
problems.
• Therefore, it is useful to turn to simulation, where virtually
anything is allowed.
• Simulation allows you to combine assumptions about uncertain
quantities and ordering policies and then play out the events as
they occur through time.
• The following example illustrates a somewhat more ambitious
ordering simulation.
• It describes a type of ordering policy, an (s, S) policy, that is
commonly used in periodic review situations.
• Example 12.8 is typical.
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Supply chain models
• One of today’s hottest areas of interest, both for
academics and business managers, is supply
chain management.
• This refers to the entire process of getting
materials from suppliers, transforming them into
finished products, and ultimately getting the
finished products to customers.
• Efficient supply chains have become a requirement
in most industries. Without them, companies
cannot compete successfully.
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Supply chain models
continued
• We consider the supply chain model in Example
12.9.
• This is a problem faced by many companies in the
fashion industry.
– When they introduce a new fashion, they are never sure
whether it will sell well or not. Therefore, a reasonable
strategy is to produce a limited amount early and see
how things go.
– If the product sells well early, they can produce more
later on—subject to capacity restrictions. If the product
does poorly early, they can cut their losses short.
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Inventory management in a
multiechelon setting
• Example 12.10 illustrates inventory management in a
multiechelon setting, that is, in a setting where
inventory is held at multiple locations.
• Although many versions of this general problem exist
in both academic articles and in real companies, we
illustrate the situation where a central warehouse holds
and distributes inventory to several retailers, each of
which has uncertain demand.
• The problem is complicated, as it usually is in real
situations, by ordering lead times, and the way
inventory should be managed is far from obvious.
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Conclusion
• We have examined a variety of inventory/ordering
models in this chapter.
• The general theme is the balance companies try to
find between competing costs.
• If they order frequent, small quantities, they keep
inventory low, but they incur large fixed ordering
costs.
• In contrast, if they order infrequent, large
quantities, they minimize ordering costs, but they
incur large holding costs.
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Conclusion continued
• The basic EOQ model and its many variations are able
to achieve the right balance between these costs.
• These EOQ models are relatively straightforward and
find many uses in today’s business world.
• However, as we introduce complications that real
companies face, such as multiple products, uncertain
demand, uncertain delivery lead times, and complex
supply chain considerations, the models can become
extremely difficult.
• In this case, simulation is often the best alternative;
sometimes, it is the only alternative.
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Summary of key management
science terms
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Summary of key management
science terms continued
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
End of Chapter 12