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P
ainting automobile bumpers can be quite complicated.
At a major bumper manufacturing facility in
Michigan, about one bumper out of four is defective
and must be rerouted and repainted. Bumpers are produced
on a single machine, one batch after another, and each
batch consists of bumpers for a particular car model and
color. Changeovers between batches are expensive and can
take a long time. The bumpers need to dry before they can
be inspected, but by then another batch is already in production
so plant managers do not know exactly how many
good bumpers they will have before starting the next batch.
Because the production yield is very unpredictable, a
sizeable stock of bumpers has to be kept on hand in a very
large storage system, with significant costs. "It was getting
unwieldy," says Chicago Booth professor John R. Birge,
whose visit to the Michigan plant motivated him and coauthors
Scott E. Grasman of Missouri University of Science
and Technology and Tava Lennon Olsen of the University
of Auckland to develop a production control system that
would allow the bumper producer, as well as other similar
manufacturing plants, to have as many bumpers available as
possible while minimizing costs.
The key is to find the right basestock level or the optimal
number of bumpers that managers plan to produce, which
they discuss in a study titled "Setting Basestock Levels in
Multi-Product Systems with Setups and Random Yield." If
managers set the basestock levels too low, then the company
may run out of bumpers to satisfy demand from car
manufacturers and the plant would have to go on overtime
and possibly hire more workers. If basestock levels are
set too high, then the company may end up with too many
bumpers that nobody wants. Both scenarios will result in
substantial costs to the company. "The idea of finding the
optimal basestock level is to set it somewhere in between
too many and too few, to get the right balance," Birge says.
Random Yield
Just like bumper plants, many modern production systems
are characterized by several uncertainties that make output
highly variable.
For instance, only about half of semiconductor chips
pass inspection after the first try, according to previous
studies. Similar to producing bumpers in several colors,
semiconductor manufacturers have to make different types of chips that go into computers, cell phones, and other digital devices and consumer appliances. Moreover, machines
that make these semiconductors tend to be very expensive,
so the chips are usually produced in different batches, one
type after another, on just one machine. As a result, it is
difficult to predict how many defective chips are in a particular
batch before production of the next batch starts.
New manufacturing facilities also typically have yields
that are very low. Output improves as more is learned
about the production process, but generally never reaches
the point where no defective items are produced. This is
because rapidly changing technologies can make a production
process obsolete even before it is well understood,
or it may not be financially justifiable to correct the yield
problem. Thus, random yield models are valuable in helping
an operation run more efficiently.
But very few studies have analyzed random yield models
when multiple products can be made on a single machine,
as with the cases of car bumpers and semiconductor chips.
Moreover, unlike other studies that make ad hoc calculations
of what the basestock levels ought to be, the methods
developed by Birge and his coauthors give more consistent
cost estimates that managers can rely on when planning
how much of each item to produce. "It allows the manufacturer
to design the sequence of products in such a way that
it can be very confident about knowing how often it would
be able to avoid using overtime and how much inventory it
will have," says Birge.
Choosing How Much to Make
The authors analyze a manufacturing system where all
products must be produced on a single machine and the
output is highly variable. Production of each item continues
until the inventory position reaches the chosen basestock
level. If a customer's order is more than what is available in
inventory, then the order is either backlogged or the sale is
lost. Because of the delay between production and inspection,
defective items cannot immediately be routed back to
the production line. Defective items are temporarily stored
in a rework storage queue until the next production cycle.
In this type of system, the best balance between too
much and too little inventory can be found by setting the
basestock levels at the point that minimizes the costs of
shortage, holding inventory, reworking defective items,
and setting up the machine before it switches over to the
next item. The method of finding
the right basestock levels will differ
depending on whether orders
can be backlogged.
If a customer wants to buy
red bumpers but the company
runs out of them, it can still take
a customer's order and deliver
the red bumpers in the future.
This backlog of orders, however,
comes with a cost: The company
has to pay a penalty for every
period that it fails to deliver the
red bumpers. Thus, in a production
system with backlogging,
finding the optimal basestock
levels involves choosing the amount of bumpers to produce
that keeps down the costs of holding inventory and setting
up for the next batch as well as penalties for orders that
have yet to be delivered.
But if backlogs are not allowed, a company will have to
turn down a customer's request for red bumpers if the order
cannot be met with what is in stock or what is currently in
production. In this case, the company loses the sale. Unlike
in backlogging, choosing the best possible basestock levels
in this type of system entails minimizing not only inventory
and setup costs but also the cost of losing sales and reworking
defective items. The expense of reworking defective items is
relevant because faulty bumpers—which cannot be
sold immediately—are repainted anyway and eventually end
up in inventory, which in turn affects the decision of how
many bumpers to produce in the next period. Rework costs
include the cost of scrapping a previous attempt at making a
bumper or the cost of getting the item ready to run through
the production process again.
The authors propose that finding the optimal basestock
levels in a system with backlogging is similar to solving the
so-called "news vendor problem," which asks how many
newspapers a vendor should put on the newsstand without
knowing what the demand will be.
On the other hand, finding the best possible basestock
levels when unmet orders lead to lost sales can be very time
consuming to calculate. To get around this difficulty, the
authors developed a method that approximates some of the
key variables that go into the optimization problem. They
find that this "heuristic" approach closely predicts the cost
of a production system with lost sales, where the difference
between the predicted and actual costs was on average
very small.
"Setting Basestock Levels in Multi-Product Systems with Setups and Random Yield." Scott E. Grasman, Tava Lennon Olsen, and John R. Birge. IIE Transactions,
2008.
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