A Near-Optimal Solution Method of Multi-Item Multi-Process Dynamic Lot Size Scheduling Problem

This paper addresses a multi-item multi-process dynamic lot size scheduling problem with general product structure and setup time. In this problem, there exist various heterogeneous decision features such as lot sizing, lot sequencing, dispatching, and so on. We present a near-optimal solution method, which we call a narrow sense Lagrangian decomposition coordination method of solving all decision features involved in this problem simultaneously without specifying or awaking to them one by one. First, splitting the planning horizon into very small time-slots, for any item on any machine at any timeslot we denote a state of processing by using a binary decision variable which takes a value of unity if it is processed, and else then zero. Second, dealing with the transition of the inventory state of each item and time transition of each setup explicitly, we formulate the problem into a multi-dimensional dynamic optimization problem with constraints. Third, paying attention to the existence of the interaction constraints, we decompose the whole problem into item-based sub problems that can be reformulated into dynamic programming of one dimension to dissolve the curse of dimensionality. At the aim of guaranteeing the decomposability, we formulate the problem by echelon inventory. The computational procedure consists of solving sub problems for given Lagrange multiplier values and of coordinating those values. Finally, we verify the presented method by using a numerical model.