Optimal Test Point Selection for Sequential Manufacturing Processes

Abstract

Consider a manufacturing process, such as the production of complex semiconductor devices, which consists of the sequential application of n possibly unreliable operations, t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , · · ·, <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> . Let c <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> be the cost incurred in performing operation t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , and let p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> be the probability that t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> will be performed successfully. Clearly one would prefer to reject immediately any item as soon as a faulty operation has been performed upon it in order to avoid the unnecessary cost of further processing that item. For this purpose, we shall assume that, immediately following each operation t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , it is possible to apply a perfectly reliable test T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , for determining whether or not the item should be rejected at that point, where the cost incurred by applying T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> depends only upon the point i of test application and the last previous point at which such a test was applied. Since the application of tests entails additional costs, careful analysis is required to determine which tests are sufficiently useful to justify that additional cost. Using a dynamic programming approach, we derive a useful and efficient algorithm which utilizes the test and operation costs, along with the operation failure probabilities, to determine a set of testing points which will result in the minimal expected manufacturing cost. We then show how it is possible to further improve upon our algorithm for the particular case in which all test costs depend only upon the point of test application.