Question

Your job is to make sure that a washing machine lasts more than 10 years, the warranty period, with probability 0.99. As it is, the mean lifetime is 10 years with standard deviation 2.5 years, and it is Gaussian distributed. You can leave the standard deviation the same for no cost. To reduce the standard deviation, it costs $100 to reduce the standard deviation to 1.5 years, or $200 to reduce it to 0.5 years. It costs $50 per year (for any positive real value of years) you wish to increase the mean of the lifetime. (For example, if you want the machine to have a mean lifetime of 11.5 years with a standard deviation of 1.5 years, the cost would be $50(11.5 − 10) = $75, plus $100 to reduce the standard deviation to 1.5, for a total cost of $175. However, this configuration would not have probability 0.99 of having a lifetime greater than 10 years.) Find the lowest cost way to achieve the engineering design goals.

Answer 1

Let mean is increased by a and standard deviation is reduced by b.

Then we want P(a>10) >= 0.99

or, P((a-10)/(2.5-b)) > 0) >=0.99

P(z>0) >=0.99

or, (a-10)/(2.5-b) >= 2.33

a-10 >= 5.825 - 2.33b

a >= 15.825 - 2.33b

The total cost is 50a + 100b

So, we min : 50a + 100b subject to a + 2.33b>= 15.825

Solving, we get : a = 0, b = 6.792 and total cost = $679.2