An item is subjected to a series of n tests (example: hardness, weight, length, etc.). Two known constants are associated with carrying out the ith test in the series. The constants are Ki (the cost per item of carrying out the ith test) and Ri (the probability of rejecting the item based on the ith test). The tests are independent in the sense that they may be run in any order, and the constants (Ki and Ri) are independent of the test order. For a given sequence of tests, an item is subjected to each test in the sequence in turn as long as the test does not reject the item (if an item is rejected by any test, no further tests are performed).

Determine the test sequence that minimizes the total expected cost of testing an item. Prove that your proposed sequence is optimal in minimizing the total expected cost of testing an item.