Example 2.3
An Ordering Decision with
Quantity Discounts
Background Information

Sam’s Bookstore, with many locations across the
United States, places orders for all of the latest books
and then distributes them to individual bookstores.

Sam’s needs a model to help it order the appropriate
number of any title.

For example, it plans to order a hot new hardback
novel, which it will sell for $30. It can purchase any
number of this book from the publisher, but due to
quantity discounts, the unit cost for all books it orders
depends on the number ordered.
Background Information -continued

Specifically
– If the number ordered is less than 1000, the unit cost is $24
– For at least 1000 copies the price is $23
– For at least 2000 copies the price is $22.25
– For at least 3000 copies the price is $21.75
– For at least 4000 copies the price is $21.30

Sam’s is very uncertain about the demand for this
book – it estimates that demand could be anywhere
from 500-4500.
Background Information -continued

Also, as with most hardback novels, this one will
eventually come out in paperback.

Therefore, if Sam’s has any hardbacks left when the
paperback comes out, it will put them on sale for $10,
at which price it believes all leftovers will be sold.

How many copies of this hardback novel should
Sam’s order from the publisher?
The Solution

First we develop a model to calculate Sam’s profit for
any order quantity and any possible demand.

Then we will perform a sensitivity analysis to see how
profit depends on these two quantities.

Finally we will indicate one possible method Sam
might use to choose the “best” order quantity.
QUANTITYDISCOUNT.XLS

The profit model shown on the next slide can be
found in this file.

Note that the OrderQuan and Demand cells are “trial”
values. We can put any values in these cells, just to
test the logic of the model.

Also note how we have used a table to indicate
quantity discounts for ordering. After entering the
inputs and trial values of order quantity and demand,
use the following steps to complete the model.
The Profit Model
Completing the Model

After entering the inputs and trial values of order
quantity and demand, proceed through the following
steps to complete the model.
– Revenues. Sam’s can sell only what it has, and it will sell
any leftovers at the sale price. Therefore enter the formulas
=MIN(OrderQuan,Demand),
=IF(OrderQuan>Demand, OrderQuan-Demand,0),
and =SoldReg*UnitPrice+SoldSale*SalePrice
in the SoldReg, SoldSale and Revenue cells.
Completing the Model -continued
– Total ordering cost. Depending on order quantity, we find
the appropriate unit cost from the unit cost table and multiply
it by the order quantity to obtain the total ordering cost. This
could be accomplished with a complex nested IF formula,
but a much better way is to use the VLOOKUP function.
Specifically, enter the formula
=VLOOKUP(OrderQuan,CostLookup,2)*OrderQuan
in the Cost cell.
– Profit. Calculate the profit with the formula =Revenue-Cost
Creating a Data Table

The next step is to create a data table for profit as a
function of the order quantity and demand. The data
table is shown here.
Creating a Data Table -continued

This table shows that profit depends heavily on both
order quanitity and demand, and how higher
demands lead to larger profits.

But is it clear which order quantity Sam’s should
select.

Remember that Sam’s has complete control over the
order quantity, but it also has no direct control over
demand.
Answering the Question

The ordering decision depends not just on which
demands are possible, but on which demands are
likely to occur.

The usual way to express this information is with a
set of probabilities that sum to 1. We show one
possible set on the next slide.

Sam’s would need to estimate these probabilies,
possibly on the basis of other similar novels it has
sold in the past.
Answering the Question -continued
Answering the Question -continued

The ones shown indicate that Sam’s believes the
most likely demands are 2000 and 2500, with other
values on either side less likely.

Now we can use these probabilities to find an
expected profit for each order quantity. This expected
profit is a weighted average of the profits in any row
in the data table, using the probabilities as the
weights. The easiest way to do this is to enter the
formula
=SUMPRODUCT(B20:J20,Probabilities) in cell B35
and copy it down to cell B43.
Answering the Question -continued

The largest of the expected profits, $12,250,
corresponds to an order quantity of 2000, so we
would recommend that Sam’s order 2000 copies of
the book.

This does not guarantee that Sam’s will make a profit
of $12,250 – the actual profit depends on the
eventual demand – but it represents a reasonable
way to proceed in the face of uncertain demand.