Zaba is a fashion retailer selling a broad line of clothing for women, men, and children. There are two main sales seasons in a year, i.e., the spring-summer season and the fall-winter season. As is typical in the fashion retailing business, the consumer demand is difficult to predict, and one of the key challenges for Zaba is to match supply with the uncertain demand. Traditionally, Zaba had to forecast demand and commit to an order quantity well in advance of the sales season, and there were no opportunities for replenishment during the season. It was frustrating when you saw a hot selling item quickly run out of stock and there was nothing you could do about it. On the other hand, it was also frustrating to see an item gradually turn into a dog (not selling) but an order commitment was already made. Luckily, as a result of the quick response movement in the retail industry, Zaba recently negotiated with its suppliers and obtained a mid-season replenishment opportunity. This question is about Zaba's mid-season replenishment decision.
To simplify the problem as much as possible, let's just consider one product in the spring-summer collection. Call it item X. The sales season goes from January to May. Two months into the sales season, Zaba can place a replenishment order for item X. That is, Zaba can place an order with a supplier for item X on March 1. The retail price is $100 per unit, and it costs Zaba $50 to purchase from the supplier. If there is unsold inventory at the end of the season, it can be salvaged in a secondary market for $30 apiece. Based on the sales in January and February, Zaba has determined that the demand from March 1 to the end of the season is normally distributed with mean 1000 and standard deviation 300. For simplicity, let us assume that no customer demand is lost between March 1 and the time the mid-season order is delivered to the store.
a) Suppose on March 1 before the ordering decision, Zaba's on-hand inventory for item X is 300 units. How much should Zaba order on March 1?
b) The supplier, because of its capacity limitations during the sales season, imposes a maximum order quantity on Zaba. That is, Zaba cannot order more than 500 units on March 1. What is Zaba's optimal mid-season order quantity?
c) Under the decision determined in b) above, what is the expected sales for the product after March 1? The expected lost sales? The expected fill rate? And finally, the expected profits? (In calculating the expected profits, assume that the 300 units remaining from the first two months of the season are costless.)
d) A second supplier comes along, who offers unlimited capacity but charges $90 per unit. Should Zaba order from this supplier? Why?