Question

Consider the following excerpts from a New York Times article: Despite its early promise … Restoration has had trouble becoming a mass-market player … What went wrong? High on its own buzz, the company expanded at breakneck speed, more than doubling the number of stores, to 94, in the year and a half after the stock offering … Company managers agree, for example, that Restoration’s original inventory system, which called for all furniture to be kept at stores instead of a central warehouse, was a disaster. Let’s look at one Restoration Hardware product, a leather chair. Average weekly sales of this chair in each store is normally distributed with mean 1.5 units and standard deviation 1.2 units. Demands are independent across stores. There is information system in place. The replenishment lead time is 12 weeks. The holding cost is $8 per unit per year, and the ordering cost is $100. a. If each store holds its own inventory, then what is the company’s average inventory if the company policy is to target a 99.25 percent in-stock probability? b. Suppose Restoration Hardware builds a central warehouse to serve the 94 stores. The lead time from the supplier to the central warehouse is 12 weeks. The lead time from the central warehouse to each store is one week. Suppose the warehouse operates with a 99 percent in-stock probability, but the stores maintain a 99.25 percent in-stock probability. If inventory at both the central warehouse and the retail stores is considered, what is Restoration’s percentage reduction in average inventory?

probability is 99.25% question 2 is 99.25% and 98%

Answer 1

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Answer 2

a. If each store holds its own inventory, then the company’s average inventory can be calculated as:

Average demand per store per week = 1.5 units Standard deviation of demand per store per week = 1.2 units Lead time = 12 weeks Service level = 99.25%

Using the formula for the reorder point with normal demand and lead time:

Reorder point = Average demand during lead time + Safety stock Safety stock = (Z-value for the service level * Standard deviation of demand during lead time)

Z-value for 99.25% service level can be found using a standard normal distribution table or calculator, and it is approximately 2.33.

Average demand during lead time = Average demand per store per week * Lead time = 1.5 units/week * 12 weeks = 18 units Standard deviation of demand during lead time = Standard deviation of demand per store per week * sqrt(Lead time) = 1.2 units/week * sqrt(12) = 4.16 units

Safety stock = 2.33 * 4.16 = 9.68 units Reorder point = 18 + 9.68 = 27.68 units

The order quantity can be calculated using the economic order quantity formula:

Order quantity = sqrt((2 * annual demand * ordering cost) / holding cost) Annual demand = 1.5 units/week * 52 weeks = 78 units Order quantity = sqrt((2 * 78 * 100) / 8) = 49.14 units, rounded up to 50 units

Therefore, the average inventory per store would be (27.68 + 50) / 2 = 38.84 units.

b. If a central warehouse is built to serve the 94 stores and the lead time from the supplier to the central warehouse is 12 weeks, the lead time from the central warehouse to each store is one week, and the warehouse operates with a 99% in-stock probability, the safety stock at the warehouse can be calculated using the formula:

Safety stock = (Z-value for 99% service level * Standard deviation of demand during lead time) Z-value for 99% service level is approximately 2.33.

Average demand during lead time = 1.5 units/week * 12 weeks = 18 units Standard deviation of demand during lead time = 1.2 units/week * sqrt(12) = 4.16 units

Safety stock = 2.33 * 4.16 = 9.68 units

Therefore, the inventory at the central warehouse should be 18 + 9.68 = 27.68 units.

To maintain a 99.25% in-stock probability at the stores, the safety stock at each store can be calculated using the same formula:

Safety stock = (Z-value for 99.25% service level * Standard deviation of demand during lead time) Z-value for 99.25% service level is approximately 2.45.

Standard deviation of demand during lead time is the same as before, 4.16 units.

Safety stock = 2.45 * 4.16 = 10.18 units

The average demand per store per week remains the same, 1.5 units. The lead time from the central warehouse to each store is one week, so the reorder point for each store should be:

Reorder point = Average demand during lead time + Safety stock = 1.5 * 1 + 10.18 = 11.68 units, rounded up to 12 units.

The order quantity can be calculated using the same formula as before:

Order quantity = sqrt((2 * annual demand *