In this thesis, we will investigate the COS problem with order-based and job-based processing approaches where jobs are produced on a single manufacturing facility (machine) to minimize the total completion time. We will first identify the basic properties of the optimal solution for the COS problem with job-based processing, develop different mixed integer programming (MILP) models to solve this COS problem optimally, and then compare the MILP models based on the size complexity. Furthermore, we will develop metaheuristic heuristic algorithm(s) to for large scale problems in which a solution cannot be obtained by the MILP model. The optimization software package GAMS will be used for solving MILP models. Computational experiments will be done to evaluate the performance of our solution approaches in terms of both solution quality and computational time, and to compare the order-based and job-based processing approaches, based on the objective function value.