To explore the properties of the generalized Gerstewitz function based on co-radiant sets and the scalarization methods for set optimization problems, this paper conducts research by leveraging the definition and properties of co-radiant sets and combining the theory of generalized Gerstewitz functions. Through the study of the continuous and decreasing properties of the generalized Gerstewitz function, this paper investigates the relationship between the solutions of set optimization problems and the corresponding scalarized problems, and obtains the corresponding scalarization results. The research results provide a new theoretical basis for solving set optimization problems and enrich the theoretical research in the fields of co-radiant sets and Gerstewitz functions.