The primary objective of this research is to construct the mathematic models in different market response functions associated with the gaming structures, and then to identify their equilibrium (or solutions) and also to explore some preference conditions for both supply chain players.In this research, four different cases of market response functions are constructed.In these different cases of market response functions, we consider the manufacturers whose polices in long-term branding investments could influence the retailers whose polices in short-term promotion efforts.Thus, this research problems become to solve the simultaneous cooperative advertising problem under different market response functions by using (1) the analytic solution approach to identify both the equilibriums of two-player of manufacturing-retailer (M-R) Nash game;its solution as a reference of the solution using swarm particle optimization-crowding distance (MOPSO-CD) or non-dominated sorting genetic algorithm (NSGA Ⅱ) and (2) by using MOPSO-CD or NSGA Ⅱ integrated the Nash game identify the equilibriums of the case of both multi-manufacturer and multi-retailer without considering the polices in long-term branding investments could be influenced each other by manufacturer firms and in short-term promotion efforts could be influenced each other by retailers in the cooperative advertising.Finally, this research will implement a real case and their numerical results will demonstrate the feasibility of the equilibrium (or solution) using MOPSO-CD or NSGA Ⅱ for solving multi-objective and multi-disciplinary optimization problems of the cooperative advertising in supply chain.