In this study, we deal with multi-objective combinatorial optimization problems because many real problems with discrete structures can be formulated for combinatorial optimization problems and there are plural objective functions in real problems. We focus on metaheuristics as optimization method for combinatorial optimization problems. In metaheuristics, how to use the search history information and the interactions among search points at making the neighborhoods and moving search points are important. We proposed a multi-point search method which is based on Tabu Search at making the neighborhoods and introducing new interactions among search points at moving search points. In this study, we analyze the method qualitatively and propose a new moving strategy based on diversification and intensification. We examine the method with the new moving strategy and validate utility of the strategy through the numerical experiment.