Lemington’s is trying to determine how many Jean
Hudson dresses to order for the spring season.
Demand for the dresses is assumed to follow a normal
distribution with mean 400 and standard deviation
100. The contract between Jean Hudson and
Lemington’s works as follows. At the beginning of the
season, Lemington’s reserves x units of capacity.
Lemington’s must take delivery for at least 0.8x
dresses and can, if desired, take delivery on up to x
dresses. Each dress sells for $160 and Hudson charges
$50 per dress. If Lemington’s does not take delivery
on all x dresses, it owes Hudson a $5 penalty for each
unit of reserved capacity that is unused. For example,
if Lemington’s orders 450 dresses and demand is for
400 dresses, Lemington’s will receive 400 dresses and
owe Jean 400($50) � 50($5). How many units of
capacity should Lemington’s reserve to maximize its
expected profit?
To maximize its expected profit, Lemington's needs to determine the optimal number of units of capacity to reserve for the Jean Hudson dresses. The goal is to balance the potential loss from unused capacity with the potential profit from selling the dresses.
To solve this problem, Lemington's can use a decision tree analysis. Here's how to calculate the optimal number of units of capacity to reserve:
1. Identify the range of possible demand values:
- The demand for the dresses is assumed to follow a normal distribution with a mean of 400 and a standard deviation of 100. The range of possible demand values can be determined by considering the mean plus or minus a certain number of standard deviations. In this case, let's consider the range from mean minus 2 standard deviations to mean plus 2 standard deviations to cover a significant portion of the demand distribution. So, the range of demand values will be [400 - (2*100), 400 + (2*100)] = [200, 600].
2. Calculate the expected profit for each possible number of units of capacity reserved:
- Start by calculating the expected demand for each possible demand value in the range [200, 600]. The expected demand can be calculated using the probability density function (PDF) of the normal distribution.
- For each possible number of units of capacity reserved (x), calculate the minimum and maximum number of dresses that Lemington's must take delivery on, which are 0.8x and x, respectively.
- Calculate the expected profit for each number of units of capacity reserved using the following formula:
- Expected Profit = (Minimum number of dresses to take delivery on * (Revenue per dress - Cost per dress)) - (Penalty cost for unused capacity)
- Revenue per dress = $160 (selling price)
- Cost per dress = $50 (charged by Jean Hudson)
- Penalty cost for unused capacity = $5 (per unit of unused capacity)
3. Determine the optimal number of units of capacity reserved:
- Compare the expected profits calculated for each possible number of units of capacity reserved.
- Select the number of units of capacity that corresponds to the highest expected profit.
By following these steps and performing the necessary calculations, Lemington's can determine the optimal number of units of capacity to reserve and maximize its expected profit.