In the field of operations research, the the Lot Sizing Problem (LSP) aims to satisfy the demand of clients while minimizing setup and inventory holding costs. The aim of this dissertation is to propose efficient tools and investigate optimization methods to solve different variants of LSP. The thesis is organized in two parts: in the first one we focus on deterministic problem while in the second one we focus on the stochastic version of the problem. In particular, in the first part, we propose a Lagrangian relaxation approach to solve the deterministic Capacitated LSP (CLSP) on multiple items, with both set up costs and set up times.