Consider a continuous operation in which raw tomatoes arrive to a processing unit in bulk. The tomatoes arrive at an average steady rate of 75 tons per hour from 8 a. m. to 1 p. m. every day. The processing begins as soon as the tomatoes start arriving, at 8 a. m. However, the processing unit can only process at a steady rate of 25 tons/hr. If need be, the processing unit can process for 24 hours a day. If tomatoes arrive when the processing unit is busy, a queue of inventory will form.
a. How many hours a day does the processing unit operate?
b. What is the maximum inventory of raw tomatoes?
c. What is the capacity utilization of the processing unit?
(Hint: This is equivalent to the fraction of a day the plant operates)
d. How long on average does a tomato stay in the queue?
(Hint: First calculate average inventory and average throughput for the entire time
the processing unit is operating. Then use little’s law to calculate CT.)

Explanation:

a) The amount of tomatoes that arrive in the 5 hours between 8 a.m. and 1 p.m each day is ...

  (75 T/h)(5 h) = 375 T

Processing at the rate of 25 T/h takes ...

  (375 T)/(25 T/h) = 15 h

The processing unit must operate 15 hours per day to process the arriving tomatoes.

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b) The arriving tomatoes are processed at the rate of 25 T/h, so the net addition to inventory from 8 a.m. to 1 p.m. is 75 -25 = 50 T/h. Over those 5 hours, an inventory accumulates in the amount of ...

  (50 T/h)(5 h) = 250 T

The maximum inventory of raw tomatoes is 250 tons.

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c) The processing unit operates 15 hours of each 24-hour day, so is operating at 15/24 = 5/8 = 62.5% of capacity.

The capacity utilization of the processing unit is 62.5%.

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d) Inventory both increases and decreases linearly, so the average inventory is half the peak inventory. Average inventory is 250/2 = 125 tons. This is processed at the rate of 25 T/h, so the amount of time the average tomato spends in inventory is ...

  (125 T)/(25 T/h) = 5 h

The average tomato stays in the queue 5 hours.

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explanation:

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