Suppose you have a set of items (rocks) that each contain a certain amount of precious metal. The amount of precious metal contained in each rock is distributed over $N(\mu_1, \sigma_1)$
You have a device that imperfectly measures the amount of precious metal in each rock with an error that is distributed over $N(\mu_2, \sigma_2)$
Our goal is to find the rock that contains the most precious metal.
The cost of choosing the wrong rock is $C = v_n - v_a $ where $v_n$ is the value of the rock with the most precious metal, and $v_a$ is the value of the rock you chose.
Furthermore, each time you perform a measurement on a rock you have the added cost of $c_m$
We can thus find the total cost $C$ from the equation: $C = (v_n - v_a) + (n*c_m)$
Where $n$ is the total number of measurements you performed.
What is the optimal strategy to minimize $C$?
Edit: I should add that $|S|$ the size of the set of rocks is known and that the true value of $v_n$ is unknown.