This study is concerned with a single-facility production system in which two kinds of products, product 1 and 2, are produced by two different processes, process 1 and 2. Product 2 is produced either by process 2 or by process 1 as by-product, while product 1 is obtained only through process 1. The repeating sequences of the two processes and their associated lot sizes are determined which minimize the total inventory related cost. A solution algorithm is developed and is illustrated with numerical examples.