Orange Juice Inc. produces and markets fruit juice. During the orange harvest se

Question

Orange Juice Inc. produces and markets fruit juice. During the orange harvest season, trucks bring oranges from the fields to the processing plant during a workday that runs from 7am to 6pm. On peak days, approximately 12 tons of oranges are trucked in every hour (i.e. inflow is continuous and constant from 7am to 6pm). Trucks dump their contents in a holding bin with a storage capacity of 6 tons. When the bin is full, incoming trucks must wait until it has sufficient available space. A conveyor moves oranges from the bins to the processing plant. The plant is configured to deal with an average harvesting day, and maximum throughput is 8 tons per hour. Assuming the oranges arrive continuously over time and all oranges must be processed the day they are picked to ensure freshness. How many hours are there between the first truck being forced to wait (i.e. inventory equals bin capacity) and the last truck being allowed to leave (i.e. the last orange is delivered from the truck into the bin)? Answer in three decimal places

Explanation / Answer

Inflow rate, Trucking rate = 12 tons/hour. (Constant flow 7am – 6 pm)

Outflow rate, Maximum throughput = 8 tons/hour.

Inventory build-up, x = Inflow rate – Outflow rate = 12 – 8 = 4 tons/hour.

Inventory at 7 a.m. = 0 (because of no overnight storage)

Inventory at 6 p.m. = 4 * 11 = 44 tons (7 a.m. - 6 p.m. = 11 hours)

Storage capacity of holding bin, S = 6 tons.

Waiting time for first truck = S / x

                                           = 44 / 6

                                                = 7.33 hours

Therefore, the first truck waits till (7 am + 7.33 hours = 2.33 p.m.

The last truck comes at 6 p.m. and enters the longest queue and hence waits for maximum time.

Depletion rate of storage bin = Maximum throughput = 8 tons/hour.

Maximum waiting time of the last truck = (Inventory at 6 pm. – S) / (Depletion rate of the bin)

                                                               = (44 – 6) / 8

                                                               = 4.75 hours (after 6 p.m.).

                                                                = 10.45 pm

So the last truck waits till 10.45 p.m.

Therefore, the amount of time between the first truck being forced to wait and the last truck being allowed to leave = Time that elapses between 2.33 pm and 10.45 p.m. = 8 hours 12 minutes.        

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