An excavation company has been contracted to supply 10 million cubic feet (cu. ft.) of dirt to a construction site. The company uses a bulldozer to collect dirt, which is then moved by a dump truck to the construction site. It costs the company $100 per hour to use the bulldozer, and it rents dump trucks at a rate of $40 per hour per truck. Each dump truck can hold 1,000 cu. ft. of dirt, and it takes an average of 12 minutes for the bulldozer to load a dump truck with dirt. It takes an average of five minutes for a fully loaded dump truck to deliver the dirt to the construction site and returns to the bulldozer for another load. Assume that the time to load a dump truck and the time for a dump truck to deliver the dirt to the construction site and return to the bulldozer follow exponential distributions. Moreover, a usual trip of the truck involves load with dirt, deliver the dirt, return to the bulldozer where it joins a queue for another load of dirt. Then, determine the most appropriate queuing model for the dirt-moving process and then use that model to determine how many dump trucks to rent in order to minimize the expected cost of fulfilling the contract.

MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
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An excavation company has been contracted to supply 10 million cubic feet (cu. ft.) of dirt
to a construction site. The company uses a bulldozer to collect dirt, which is then moved by a
dump truck to the construction site. It costs the company $100 per hour to use the bulldozer,
and it rents dump trucks at a rate of $40 per hour per truck. Each dump truck can hold 1,000
cu. ft. of dirt, and it takes an average of 12 minutes for the bulldozer to load a dump truck
with dirt. It takes an average of five minutes for a fully loaded dump truck to deliver the dirt
to the construction site and returns to the bulldozer for another load.
Assume that the time to load a dump truck and the time for a dump truck to deliver the dirt
to the construction site and return to the bulldozer follow exponential distributions. Moreover,
a usual trip of the truck involves load with dirt, deliver the dirt, return to the bulldozer where
it joins a queue for another load of dirt.
Then, determine the most appropriate queuing model for the dirt-moving process and then
use that model to determine how many dump trucks to rent in order to minimize the expected
cost of fulfilling the contract.

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