Optimizing Inventory Management and Bagel Production for CQ-EMBA Coffee Shop

QUESTION

The CQ-EMBA Coffee Shop is a small coffee shop with branches only in Ithaca and Kingston. They serve hungry EMBA students as well as other townspeople who happen to drop in their stores. The coffee shop carries Gimme!’s freshly roasted coffee. The Kingston branch has the coffee shipped over from Ithaca. There is a fixed fee of $30 for each shipment of coffee. Gimme! charges the coffee shop $12/pound for the coffee and the coffee shop uses a 20% interest rate for holding cost. The Kingston branch uses 20 pounds of coffee per day with a standard deviation of 4 pounds per day. The lead time for new coffee to arrive to the Kingston branch once it is ordered is 4 days. Since the EMBA students cannot live without coffee, the coffee shop would like to maintain a service level of 99.9%.

a) Currently, the coffee shop orders 200 pounds of coffee when the amount of coffee in inventory goes down to 100 pounds. What is the annual fixed ordering cost and holding cost if they continue to use this policy?

b) Under the current policy, what is the average time spent by a pound of coffee at the Kingston branch?

c) One of the EMBA students suggested that the coffee shop can improve its inventory policy by using an optimal order quantity and reorder point. Compute the optimal order quantity and reorder point the coffee shop should use.

d) The Ithaca branch also sells fresh raisin bagels. They have found the number of raisin bagels demanded by their customers per day is uniform between 1 and 50 per day (i.e., they are equally likely to sell any number of bagels between 1 and 50). It costs 35 cents to make a raisin bagel and a raisin bagel is sold for $1.00. Any bagels not sold the day they are made are discounted to 20 cents, and sold as “day-old” bagels. Assume that the demand for day-old bagels is separate from the demand for fresh bagels and that all day-old bagels are always sold. How many raisin bagels should the coffee shop make each morning?

ANSWER

To calculate the annual fixed ordering cost and holding cost under the current policy, we first need to determine the annual ordering quantity and the reorder point. The coffee shop orders 200 pounds of coffee when the inventory level reaches 100 pounds.

Annual Ordering Quantity (Q): Q = Order quantity = 200 pounds Annual Demand (D): The demand for coffee is not mentioned directly, but we can calculate it based on the daily usage and the number of working days in a year. Daily usage = 20 pounds Working days in a year (assuming 365 days) = 365 days D = Daily usage x Working days = 20 pounds/day x 365 days/year = 7,300 pounds/year

Reorder Point (ROP): ROP is the level at which a new order should be placed. In this case, it’s when the inventory reaches 100 pounds.

Now, we can calculate the relevant costs:

1. Annual Ordering Cost (CO): CO = (D / Q) x S Where S is the fixed cost per order, which is $30. CO = (7,300 pounds / 200 pounds) x $30 = $1,095
2. Annual Holding Cost (CH): CH = (Q / 2) x H Where H is the holding cost per pound, which is determined by the interest rate. The holding cost rate is 20%. CH = (200 pounds / 2) x 0.20 x $12/pound = $480

Now, add the annual ordering cost and annual holding cost to get the total annual cost:

Total Annual Cost = CO + CH = $1,095 + $480 = $1,575

b) To calculate the average time spent by a pound of coffee at the Kingston branch, we can use the Economic Order Quantity (EOQ) formula:

EOQ = sqrt((2 * D * S) / H)

Where: D = Annual demand (7,300 pounds) S = Ordering cost per order ($30) H = Holding cost per pound per year (20% of $12, which is $2.40)

EOQ = sqrt((2 * 7,300 * 30) / 2.40) = sqrt(219,000) / 2.40 ≈ 230 pounds

Now, we can calculate the average time spent by a pound of coffee at the Kingston branch using Little’s Law:

Average Inventory (I) = EOQ / 2 = 230 pounds / 2 = 115 pounds

Average Time spent (T) = I / D = 115 pounds / 7,300 pounds/year ≈ 0.0158 years

c) To compute the optimal order quantity (EOQ) and reorder point, we already calculated EOQ in part b, which is approximately 230 pounds.

Reorder Point (ROP): ROP is the level at which a new order should be placed to maintain a 99.9% service level. To do this, we need to calculate the safety stock:

Safety Stock (SS) = Z * σL Where: Z is the Z-score corresponding to the desired service level (99.9% = 3.0902) σL is the standard deviation of demand during lead time (4 pounds)

SS = 3.0902 * 4 pounds = 12.36 pounds

ROP = D * L + SS Where L is the lead time (4 days).

ROP = 7,300 pounds/year * (4 days / 365 days/year) + 12.36 pounds ≈ 79 pounds

So, the optimal order quantity is approximately 230 pounds, and the reorder point is approximately 79 pounds.

d) To determine how many raisin bagels the coffee shop should make each morning, we need to consider the demand distribution and cost structure.

The demand for raisin bagels is uniform between 1 and 50 per day, meaning any quantity between 1 and 50 is equally likely. To maximize profit, the coffee shop should make the quantity that maximizes expected profit.

Expected profit per bagel sold: Profit per fresh bagel = $1.00 – $0.35 = $0.65 Profit per day-old bagel = $0.20

To calculate expected profit, we’ll consider each quantity from 1 to 50:

Expected Profit = Σ(Probability of selling x Profit per bagel)

Let’s calculate it:

Expected Profit = [Σ(1/50 * $0.65) from 1 to 50] + [Σ(1/50 * $0.20) from 1 to 50]

Expected Profit ≈ ($0.65/50) * [1 + 2 + 3 + … + 50] + ($0.20/50) * [1 + 2 + 3 + … + 50]

Now, calculate the sum of integers from 1 to 50, which is (50 * 51) / 2 = 1275:

Expected Profit ≈ ($0.65/50) * 1275 + ($0.20/50) * 1275 Expected Profit ≈ ($8.44) + ($2.55) Expected Profit ≈ $10.99

So, to maximize expected profit, the coffee shop should make approximately 11 raisin bagels each morning.

In conclusion, by implementing an optimal order quantity and reorder point, the coffee shop can improve its inventory management and reduce costs. Additionally, for raisin bagels, making around 11 each morning would maximize expected profit considering the demand distribution and cost structure.