Production planning using mixed lots

We consider the production planning of a manufacturing system capable of producing several types of part. The demands are known at the end of each elementary period (at the end of each day or week for instance). The objective function (i.e. the criterion) is the sum of inventory, backlogging and set-up costs on a finite horizon. We use the concept of lot to simplify management decisions: instead of launching in production parts type by type, we launch lots. A lot is a mix of parts of different types. The drawback is the definition of the manufacturing times of the lots. We consider that the manufacturing time of a lot is its manufacturing time in steady state: it is a way to associate a manufacturing time to each lot, independently from the work-in-process and the set-up times. As a consequence, we have to launch identical lots in series to reduce the error made by introducing this hypothesis. In this paper, we propose a heuristic algorithm and a branch-and-bound approach to solve this problem. Numerical examples are provided