Recently, various Artificial Intelligence (AI) based optimization metaheuristics are proposed and applied for a variety of problems. Cohort Intelligence (CI) algorithm is a socio inspired optimization technique which is successfully applied for solving several unconstrained & constrained real-world problems from the domains such as design, manufacturing, supply chain, healthcare, etc. Generally, real-world problems are constrained in nature. Even though most of the Evolutionary Algorithms (EAs) can efficiently solve unconstrained problems, their performance degenerates when the constraints are involved. In this paper, two novel constraint handling approaches based on modulus and hyperbolic tangent probability distributions are proposed. Constrained CI algorithm with constraint handling approaches based on triangular, modulus and hyperbolic tangent is presented and applied for optimizing advanced manufacturing processes such as Water Jet Machining (WJM), Abrasive Jet Machining (AJM), Ultrasonic Machining (USM) and Grinding process. The solutions obtained using proposed CI algorithm are compared with contemporary algorithms such as Genetic Algorithm, Simulated Annealing, Teaching Learning Based Optimization, etc. The proposed approaches achieved 2%-127% maximization of material removal rate satisfying hard constraints. As compared to the GA, CI with Hyperbolic tangent probability distribution achieved 15%, 2%, 2%, 127%, and 4% improvement in MRR for AJMB, AJMD, WJM, USM, and Grinding processes, respectively contributing to the productivity improvement. The contributions in this paper have opened several avenues for further applicability of the proposed constraint handling approaches for solving complex constrained problems.