A Company's Manufacturing Process Uses 500 Gallons of Water at a Time

question 82

Essay

A company's manufacturing process uses 500 gallons of water at a time. A "scrubbing" machine then removes most of a chemical pollutant before pumping the water into a nearby lake. Legally the treated water should contain no more than 80 parts per million of the chemical, but the machine isn't perfect and it is costly to operate. Since there's a fine if the discharged water exceeds the legal maximum, the company sets the machine to attain an average of 75 ppm for the batches of water treated. They believe the machine's output can be described by a Normal model with standard deviation 4.2 ppm. (SHOW WORK.)
a. What percent of the batches of water discharged exceed the 80ppm standard?
b. The company's lawyers insist that they have not more than 2% of the water over the
limit. To what mean value should the company set the scrubbing machine? Assume the standard deviation does not change.
c. Because achieving a mean that low would raise the costs too much, they decide to leave the mean set at 75 ppm and try to reduce the standard deviation to achieve the "only 2% over" goal. Find the new standard deviation needed.
d. Explain what achieving a smaller standard deviation means in this context.

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Definitions:

Perishable Inventory

Items in inventory that have a limited shelf life and therefore require more dynamic and sensitive handling, management, and forecasting strategies to minimize losses due to spoilage.

Disaggregation

The process of breaking down a complex system or data set into smaller, more manageable parts.

Master Production Schedule

A detailed plan that outlines what a business needs to produce, how much, and when, over a certain period.

Aggregate Plan

An aggregate plan is a strategy for meeting forecasted demand by adjusting production rates, workforce levels, inventory levels, and other operational variables in the short to medium term.