1. (6 points) Daily demand for the ice creams at I-Scream parlor is normally distributed with a mean of 160 quarts and a standard deviation of 100 quarts. The owner has the ice cream supplied by a wholesaler who charges $2.20 per quart. The ice cream sells for $4 per quart. The wholesaler charges a $400 delivery charge independent of order size. It takes 4 days for an order to be supplied. The opportunity cost of capital to I-Scream is estimated to be 25% per year. Assume 360 days in the year.
(a) The optimal order size of each order is (in quarts):
D = 160 x 360 quarts per year, S = $400, h = 0.25 $ per year, C = $2.20, H = hC = 0.25 x 2.20 = $0.55 per quart per year
EOQ = sqrt ( ( 2 D S)/H) = sqrt ( (2 x 160 x 360 x 400)/0.55) = 9153
(b) The owner would like to ensure no stock-outs in 95% of the cycles (i.e., the service level is 95%). The safety stock the store should have is (in quarts):
SS = z σ sqrt(L) = 1.645 x 100 x sqrt(4) = 329
(c) Currently the owner orders 4000 quarts of ice cream when they have 1680 quarts on hand. Compute the total annual inventory cost including the cost of holding the safety stock.
Ordering cost = D/Q S = (160 x 360)/4000 x 400 = $5,760/year
Cycle inventory = Q/2 = 4000/2 = 2000;
safety stock = ROP – mean leadtime demand = 1680 – 160 x 4 = 1040
Total inventory = 2000 + 1040 = 3040 (i.e., cycle inventory + safety stock)
Annual inventory holding cost = 3040 x 0.55 = $1672/year
Total annual inventory cost = 5,760 + 1672 = $7,432 per year
2. [6 points] A small cafeteria serves four customers per hour. There is one server who takes twelve minutes to serve a customer. State what type of queueing system this is and compute the average number of customers in queue. If the cost of waiting to be served is $15 per hour to a customer compute the expected cost of customers waiting in line. What is the 99% confidence level for the number in queue?
This is a single server queue. Arrival rate = 4/hour. Service rate = 5 per hour. Utilization = 4/5 = 80%. Lq from table = 3.2 customer. Cost per hour = $15 x Lq = $48 per hour. 99% confidence level is 19 customers.
3. [4 points] United Parcel Service (UPS) wants delivery to the Stern School of Business in New York City to be guaranteed to take place between 1pm and 2pm. The UPS truck leaves the UPS warehouse at 11:30am. Due to traffic congestion and multiple drop-off points, the time to reach the Stern is normally distributed with an average of 2 hours and a standard deviation of 1 hour.
a) What type of control chart will you use to monitor the timeliness of deliveries? Explain.
• p chart
• x-bar chart
• x-bar and R chart
• none of above
We can observe all deliveries in a week and see if they are on time or not. So we can use a p chart (fraction of deliveries not on time in a week).
Or, we can compute the exact time of delivery and plot the average and range of the delivery times for each week. Thus, use a x-bar or R chart.
Either answer with explanation gets you full credit.
b) Based on the given information is this a six sigma process? Explain and estimate the sigma level of this process (approximately). Give suggestions for making it into a six sigma process.
Definitely not a six sigma process as the mean delivery time is 11:30 + 2 = 1:30 which is right in the middle of 1 and 2 pm but standard deviation = 1. So Cpk = min ( (1:30 – 1)/3, (2-1:30)/3) = 0.166. From table this is a two sigma process which produces 308,770 defects per million opportunities!
Either reduce the standard deviation or customer to move the tolerance limits ( from 1 to 2 pm to 1:30 +/- six sigma = 1:30 +/- 6 = 7:30 am to 7:30 pm. 4. [2 points] Sport Obermeyer manufactures Parkas using an assembly line. It owns two factories, one in Mexico and the other in China. The workers in the Mexican factory are...