The residents of Bucktown, Illinois, place their trash at the curb each Wednesda

Question

The residents of Bucktown, Illinois, place their trash at the curb each Wednesday morning to be picked up by Municipal Crews. Experience shows the total amount of trash put out has a normal distribution with a mean of 35 tons and a standard deviation of 9 tons. Crews of full-time city employees assigned to trash collection collect trash. Each group can collect five tons of trash per working day. The city has plenty of trucks of the kind used for trash collection. The marginal cost of operating one trash collection crew for one working day, including both personnel-related costs and truck-related costs, is reckoned at $625. Whatever trash remains at the end of the workday must be collected that evening by an outside contractor who charges $650 per ton. How many crews should the city assigned to trash collection? Treat the number of Crews as a continuous variable.

Explanation / Answer

Here, we need to calculate what is the service level that is appropriate as pr the cost of understocking and overstocking the trash.

Here, cost of overstocking, Co = $625/5 = $125. Here, we divided by 5 since $625 is the cost for 5 tons that can be collected by a crew.

Cost of understocking, Cu = $650/ton - $125/ton = $525/ton

Service Level (SL) = Cu/(Cu + Co) = 525/(525 + 125) = 0.807692

Now, optimum number of crews to be assigned = (Mean + Z*Std_Dev)/5

Mean = 35, Std_Dev = 9, Z= NORMSINV(Service Level) = NORMSINV(0.807692) = 0.869424

So, optimum no. of crews to be assigned = (35 + 0.869424*9)/5 = 42.82481/5 = 8.56, which means 9 crews.

So, 9 no. of crews should be assigned to trash collection.

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