CASE: Swimsuit Production
Consider a company that designs, produces, and sells summer fashion items
such as swimsuits. About six months before summer, the company must
commit itself to specific production quantities for all its products. Since there
is no clear indication of how the market will respond to the new designs, the
company needs to use various tools to predict demand for each design, and
plan production and supply accordingly.
In this setting the trade-offs are clear: overestimating customer demand will
result in unsold inventory while underestimating customer demand will lead
to inventory stockouts and loss of potential customers. To assist
management in these decisions, the marketing department uses historical
data from the last five years, current economic conditions, and other factors
to construct a probabilistic forecast of the demand for swimsuits. They have
identified several possible scenarios for sales in the coming season, based
on such factors as possible weather patterns and competitors’ behavior, and
assigned each a probability, or chance of occurring. For example, the
marketing department believes that a scenario that leads to 8,000 unit sales
has an 11 percent chance of happening; other scenarios leading to different
sales levels have different probabilities of occurring. These scenarios are
illustrated in Figure 3-3. This probabilistic forecast suggests that average
demand is about 13,000 units, but there is a probability that demand will be
either larger than average or smaller than average. Additional information
available to the manufacturer includes:
To start production, the manufacturer has to invest $100,000 independent
of the amount produced. We refer to this cost as the fixed production cost.
•
•
The variable production cost per unit equals $80.
•
During the summer season, the selling price of a swimsuit is $125 per unit.
Any swimsuit not sold during the summer season is sold to a discount store
for $20. We refer to this value as the salvage value.
•
To identify the best production quantity, the firm needs to understand the
relationship between the production quantity, customer demand, and profit.
Suppose the manufacturer produces 10,000 units while demand ends at
12,000 swimsuits. It is easily verified that profit equals revenue from
summer sales minus the variable production cost minus the fixed production
cost. That is:
Profit = 125(10,000) − 80(10,000) − 100,000 = 350,000
On the other hand, if the company produces 10,000 swimsuits and demand
is only 8,000 units, profit equals revenue from summer sales plus salvage
value minus the variable production cost minus the fixed production cost.
That is:
Profit = 125(8,000) + 20(2,000) − 80(10,000) − 100,000 = 140,000
Notice that the probability that demand is 8,000 units is 11 percent while
the probability that demand is 12,000 units is 27 percent. Thus, producing
10,000 swimsuits leads to a profit of $350,000 with probability of 27 percent
and a profit of $140,000 with probability of 11 percent. In similar fashion,
one can calculate the profit associated with each scenario given that the
manufacturer produces 10,000 swimsuits. This allows us to determine the
expected (or average) profit associated with producing 10,000 units. This
expected profit is the total profit of all the scenarios weighted by the
probability that each scenario will occur. We would, of course, like to find the
order quantity that maximizes average profit. What is the relationship
between the optimal production quantity and average demand, which, in
this example, is 13,000 units?
Should the optimal order quantity be equal to, more than, or less than the
average demand? To answer these questions, we evaluate the marginal
profit and marginal cost of producing an additional swimsuit. If this swimsuit
is sold during the summer season, then the marginal profit is the difference
between the selling price per unit and the variable production cost per unit,
which is equal to $45. If the additional swimsuit is not sold during the
summer season, the marginal cost is the difference between the variable
production cost and the salvage value per unit, which is equal to $60. Thus,
the cost of not selling this additional swimsuit during the summer season is
larger than the profit obtained from selling it during the season. Hence, the
best production quantity will in general be less than the average demand.
Figure 3-4 plots the average profit as a function of the production quantity. It
shows that the optimal production quantity, or the quantity that maximizes
average profit, is about 12,000. It also indicates that producing 9,000 units
or producing 16,000 units will lead to about the same average profit of
$294,000. If, for some reason, we had to choose between producing 9,000
units and 16,000 units, which one should we choose?
To answer this question, we need to better understand the risk associated
with certain decisions. For this purpose, we construct a frequency histogram
(see Figure 3-5) that provides information about potential profit for the two
given production quantities, 9,000 units and 16,000 units. For instance,
consider profit when the production quantity is 16,000 units. The graph
shows that the distribution of profit is not symmetrical. Losses of $220,000
happen
about
11
percent
of the time while profits of at least $410,000 happen 50 percent of the time.
On the other hand, a frequency histogram of the profit when the production
quantity is 9,000 units shows that the distribution has only two possible
outcomes. Profit is either $200,000 with probability of about 11 percent, or
$305,000 with probability of about 89 percent.
Thus, while producing 16,000 units has the same average profit as
producing 9,000 units, the possible risk on the one hand, and possible
reward on the other hand, increases as we increase the production size. To
summarize:
The optimal order quantity is not necessarily equal to forecast, or average,
demand.
Indeed,
the optimal quantity depends on the relationship between marginal profit
achieved from selling an additional unit and marginal cost. More
importantly, the fixed cost has no impact on the production quantity, only
on the decision whether to produce or not. Thus, given a decision to
produce, the production quantity is the same independently of the fixed
production cost.
•
As the order quantity increases, average profit typically increases until the
production quantity reaches a certain value, after which the average profit
starts decreasing.
•
•
As we increase the production quantity, the risk — that is, the probability
of large losses —always increases. At the same time, the probability of large
gains also increases. This is the risk/reward trade-off.
THE EFFECT OF INITIAL INVENTORY
Suppose now that the swimsuit under consideration is a model produced
last year, and that the manufacturer has an initial inventory of 5,000 units.
Assuming that demand for this model follows the same pattern of scenarios
as before, should the manufacturer start production, and if so, how many
swimsuits should be produced?
If the manufacturer does not produce any additional swimsuits, no more
than 5,000 units can be sold and no additional fixed cost will be incurred.
However, if the manufacturer decides to produce, a fixed production cost is
charged independent of the amount produced.
To address this issue, consider Figure 3-6, in which the solid line represents
average profit excluding fixed production cost while the dotted curve
represents average profit including the fixed production cost. Notice that the
dotted curve is identical to the curve in Figure 3-4 while the solid line is
above the dotted line for every production quantity; the difference between
the two lines is the fixed production cost.
Notice also that if nothing is produced, average profit can be obtained from
the solid line in Figure 3-6 and is equal to
225,000 (from the figure) + 5,000 × 80 = 625,000
where the last component is the variable production cost already included in
the $225,000.
On the other hand, if the manufacturer decides to produce, it is clear that
production should increase inventory from 5,000 units to 12,000 units. Thus,
average profit in this case is obtained from the dotted line and is equal to
371,000 (from the figure) + 5,000 × 80 = 771,000
Since the average profit associated with increasing inventory to 12,000 units
is larger than the average profit associated with not producing anything, the
optimal policy is to produce 7,000 = 12,000 − 5,000 units.
Consider now the case in which initial inventory is 10,000 units. Following
the same analysis used before, it is easy to see that there is no need to
produce anything because the average profit associated with an initial
inventory of 10,000 is larger than what we would achieve if we produce to
increase inventory to 12,000 units. This is true because if we do not
produce, we do not pay any fixed cost; if we produce, we need to pay a fixed
cost
independent
of
the
amount
produced.
Thus, if we produce, the most we can make on average is a profit of
$375,000. This is the same average profit that we will have if our initial
inventory is about 8,500 units and we decide not to produce anything.
Hence, if our initial inventory is below 8,500 units, we produce to raise the
inventory level to 12,000 units. On the other hand, if initial inventory is at
least 8,500 units, we should not produce anything.