Question

(Cranberries) International Cranberry Uncooperative (ICU) is a competitor to the National Cranberry Cooperative (NCC). At ICU, barrels of cranberries arrive on trucks at a rate of 150 barrels per hour and are processed continuously at a rate of 100 barrels per hour. Trucks arrive at a uniform rate over eight hours, from 6:00 a.m. until 2:00 p.m. Assume the trucks are sufficiently small so that the delivery of cranberries can be treated as a continuous inflow. The first truck arrives at 6:00 a.m. and unloads immediately, so processing begins at 6:00 a.m. The bins at ICU can hold up to 200 barrels of cranberries before overflowing. If a truck arrives and the bins are full, the truck must wait until there is room in the bins. a. What is the maximum number of barrels of cranberries that are waiting on the trucks at any given time? [3.3] b. At what time do the trucks stop waiting? [3.3] c. At what time do the bins become empty? [3.3] d. ICU is considering using seasonal workers in addition to their regular workforce to help with the processing of cranberries. When the seasonal workers are working, the processing rate increases to 125 barrels per hour. The seasonal workers would start working at 10:00 a.m. and finish working when the trucks stop waiting. At what time would ICU finish processing the cranberries using these seasonal workers? [3.3]

Answer 1

1.

Arrival of barrels per hour = 150

Processing of barrels per hour = 100

Buildup of barrels per hour = (150 - 100) = 50

Given that the truck arrives for 8 hours.

Maximum buildup will occur after 8 hours.

Buildup of barrels after 8 hours = 50 × 8 = 400 barrels

​​​​​​Number of barrels that can be stored in the bin = 200

Maximum number of barrels waiting on trucks after 8 hours = 400 - 200 = 200

2.

Maximum number of barrels waiting on truck after 8 hours, i.e., 2 pm = 200 (as calculated in above part)

​​​​​​Processing rate of barrels = 100 per hour

Time it will take to process 200 barrels = 200/100

Time it will take to proces 200 barrels = 2 hours

Therefore, truck will wait for 2 hours after 2 p.m. which means that truck will wait till 4 p.m.

3.

Total build up of barrels after 2 pm is 400 out of which 200 barrels are held in the bin and 200 are stored in the truck.

200 barrels of truck take 2 hours to process and the rest 200 barrels of truck will take another 2 hours to process.

Therefore bins will get empty at 6 p.m.

4.

Only Regular workers process the barrels till 10 am at a rate of 100 barrels per hour. After 10 am, seasonal workers join regular workers to process barrels at the rate of 125 per hour.

Arrival rate of barrels = 150 barrels per hour

Trucks arrive for 8 hours.

Total number of barrels that arrive in 8 hours = 150 × 8

Total number of barrels that arrive in 8 hours = 1200

FOR TIME FROM 6 A.M. TO 10 A.M.

Processing Rate = 100 barrels per hour

Number of barrels that can be processed in 4 hours = 400

Number of barrels left = 1200 - 400 = 800

​​​​​​FOR TIME AFTER 10 A.M.

Processing Rate = 125 barrels per hour

Left over barrels = 800

Time to process 800 barrels = 800/125

Time to process 800 barrels = 6.4 hours

Or

Time to process 800 barrels = 6 hour 24 minutes

Therefore processing will finish 6 hours and 24 minutes after 10 A.M.

Processing will finish at 4:24 P.M.