In this thesis, a single-level, multi-item capacitated lot sizing problem with setup carryover, setup splitting and backlogging is investigated. This problem is typically used in the tactical and operational planning stage, determining the optimal production quantities and sequencing for all the products in the planning horizon. Although the capacitated lot sizing problems have been investigated with many different features from researchers, the simultaneous consideration of setup carryover and setup splitting is relatively new. This consideration is beneficial to reduce costs and produce feasible production schedule. Setup carryover allows the production setup to be continued between two adjacent periods without incurring extra setup costs and setup times. Setup splitting permits the setup to be partially finished in one period and continued in the next period, utilizing the capacity more efficiently and remove infeasibility of production schedule.

The main approaches are that first the simple plant location formulation is adopted to reformulate the original model. Furthermore, an extended formulation by redefining the idle period constraints is developed to make the formulation tighter. Then for the purpose of evaluating the solution quality from heuristic, three types of valid inequalities are added to the model. A fix-and-optimize heuristic with two-stage product decomposition and period decomposition strategies is proposed to solve the formulation. This generic heuristic solves a small portion of binary variables and all the continuous variables rapidly in each subproblem. In addition, the case with demand backlogging is also incorporated to demonstrate that making additional assumptions to the basic formulation does not require to completely altering the heuristic.

The contribution of this thesis includes several aspects: the computational results show the capability, flexibility and effectiveness of the approaches. The average optimality gap is 6% for data without backlogging and 8% for data with backlogging, respectively. In addition, when backlogging is not allowed, the performance of fix-and-optimize heuristic is stable regardless of period length. This gives advantage of using such approach to plan longer production schedule. Furthermore, the performance of the proposed solution approaches is analyzed so that later research on similar topics could compare the result with different solution strategies.