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A fashion retailer in Santa Barbara, California, presents a new designer dress at one of the “by invitation only” fashion shows. After the show, the dress will be sold at the company’s boutique store for $10,000 apiece. Demand at the boutique is limited due to the short time the dress remains fashionable and is estimated to be normal with mean 70 and standard deviation 40. There were only 100 dresses produced to maintain exclusivity and high price. It is the company’s policy that all unsold merchandise is destroyed. a) How many dresses remain unsold on average at the end of the season? The z-statistic = (100 − 70)/40 = 0.75.
Expected lost sales = 40 × L(z) = 40 × 0.1312 = 5.248.
Expected sales = 70 − 5.248 = 64.752.
Expected leftover inventory = 100 − 64.752 = 35.248.
b) What is the retailer’s expected revenue?
Expected revenue = $10,000 × 64.752 = $647,520.
c) Fashion companies often sell a portion of new merchandise at exhibitions for a discount while the product is still “fresh” in the minds of the viewers. The company decides to increase revenues by selling a certain number of dresses at a greatly discounted price of $6,000 during the show. Later, remaining dresses will be available at the boutique store for a normal price of $10,000. Typically, all dresses offered at the show get sold, which, of course, decreases demand at the store: it is now normal with mean 40 and standard deviation 25. How many dresses should be sold at the show?
Underprotecting in this case means the underage cost is $10,000 − $6,000 = $4,000 Overprotecting means a loss of $6,000 in revenue.
The critical ratio is 4,000/(6,000 + 4,000) = 0.4.
After looking to the Standard Normal Distribution Function Table, we see
Φ(−0.25) = 0.4013 (closest), so we choose z = −0.25. Q = 40 − 0.25 × 25 = 33.75. It means we protect 34 dresses for sales at the boutique, which means we sell 100 − 34 = 66 dresses at the show.
d) Given your decision in part c, what is expected revenue?
If 34 dresses are sent to the boutique, then expected lost sales are σ × L(z) = 25 × L(−0.25) = 25 × 0.5363 = 13.41. So, expected sales = 40 − 13.41 = 26.59.
It means revenue = 26.59 × $10,000 + (100 − 34) × 6,000 = $661,900.
e) Given your decision in part c, how many dresses are expected to remain unsold?
Based on conclusions of part d, expected sales are 26.59, which means expected leftover inventory is 34 − 26.59 = 7.41 dresses.