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 14 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 9 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)? Submit answer with three decimal places (i.e. do not round too early in your calculations). Hint: Construct an inventory buildup diagram. Hint2: The last truck is allowed to leave after 6pm; the truck needs to wait until all of his oranges can be placed into the bin. Hint3: The waiting starts at the instant the inventory of oranges in the bins = the bin size. Hint4: Do NOT model individual trucks. Just assume oranges flow continuously from trucks into the bin.

Explanation / Answer

From 7am-6pm

Because inflows exceed outflows, inventory will build up at a rate of

R = 14 tons/hr – 9 tons/hr = 5 tons/hr

Because oranges cannot be stored overnight, let’s start with an empty plant so that inventory at 7am is zero:

I7 (7 am) = 0 units

Because inventory builds up linearly at 5 tons/hr, the inventory at 6pm is

I6 (6pm) = 5 tons/hr * 11 hr = 55 tons

After 6pm, no more oranges come in

Yet processing continues at 9 tons/hr until the plant is empty.

Thus, inflows is less than outflows so that inventory is depleted at a rate of

R = 0 – 9 tons/hr = - 9 tons/hr

We have I(6pm) = 55 tons

Inventory depletes linearly from that level at a rate of -9 tons/hr

To empty the plant,

55 tons = 9 tons/hr x t

or

t = 55/9 hr = 6.11 hr = 6 hr 7 min.

Thus, the plant must operate until 6pm + 6hr 7 min = 00:07 AM

b) At what point during the day must a truck first wait before unloading into the storage bin?

Inventory builds up in the bins. When the bin is full, then the trucks must wait. This happens at:

5 tons/hr x t = 6tons,

The first truck will wait after t = 6/5 hr = 1.2 hr, which is at 6 am + 1hr 12 mins or 7 hr 12mins.

Thus, the first truck waits at 7:12 AM

c) What is the maximum amount of time that a truck must wait?

The last truck that arrives (at 6pm) joins the longest queue, and thus will wait the longest. It will be able to start dumping its contents in the bins when the bins start depleting. This is at

55 tons – (9 tons/hr) x t = 6 tons of bin

t = (55 – 6)/9 = 5.44 hours

Thus, last truck has to wait more 5.44 hours after 7 PM

The maximum truck waiting time is therefore 5 hours 27 minutes, and that is 00:27 AM.

The trucks will start waiting on 07:12 AM to 00:27 AM

Hours of waiting = 16 hours 45 minutes

Waiting hours 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) = 16 hours 45 minutes.

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