HVAC Manufacturing produces parts and materials for the heating, ventilation, and air conditioning industry. One of its facilities produces metal ductwork in various sizes for the home construction market. One particular product is 6-inch diameter round metal ducting. It is a simple product, but the diameter of the finished ducting is critical. If it is too small or large, contractors will have difficulty fitting the ducting into other parts of the system. The target diameter is 6 inches exactly, with an acceptable tolerance of ±.03 inches. Anything produced outside of specifications is considered defective. The line supervisor for this product has datashowing that the actual diameter of the finished product is 5.99 inches with a standard deviation of .01 inches.a. What is the current capability index of this process? What is the probability of producing a defective unit in this process?b. The line supervisor thinks he will be able to adjust the process so that the mean diameter of output is the same as the target diameter, without any change in the process variation. What would the capability index be if he is successful? What would be the probability of producing a defective unit in this adjusted process?c. Through better training of employees and investment in equipment upgrades, the company could produce output with a mean diameter equal to the target and a standard deviation of .005 inches. What would the capability index be if this were to happen? What would be the probability of producing a defective unit in this case?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
HVAC Manufacturing produces parts and materials for the heating, ventilation, and air conditioning industry. One of its facilities produces metal ductwork in various sizes for the home construction market. One particular product is 6-inch diameter round metal ducting. It is a simple product, but the diameter of the finished ducting is critical. If it is too small or large, contractors will have difficulty fitting the ducting into other parts of the system. The target diameter is 6 inches exactly, with an acceptable tolerance of ±.03 inches. Anything produced outside of specifications is considered defective. The line supervisor for this product has data
showing that the actual diameter of the finished product is 5.99 inches with a standard deviation of .01 inches.
a. What is the current capability index of this process? What is the
b. The line supervisor thinks he will be able to adjust the process so that the mean diameter of output is the same as the target diameter, without any change in the process variation. What would the capability index be if he is successful? What would be the probability of producing a defective unit in this adjusted process?
c. Through better training of employees and investment in equipment upgrades, the company could produce output with a mean diameter equal to the target and a standard deviation of .005 inches. What would the capability index be if this were to happen? What would be the probability of producing a defective unit in this case?
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