Energy-efficient production scheduling research has received much attention because of the massive energy consumption of the manufacturing process. In this article, we study an energy-efficient job-shop scheduling problem with sequence-dependent setup time, aiming to minimize the makespan, total tardiness and total energy consumption simultaneously. To effectively evaluate and select solutions for a multiobjective optimization problem of this nature, a novel fitness evaluation mechanism (FEM) based on fuzzy relative entropy (FRE) is developed. FRE coefficients are calculated and used to evaluate the solutions. A multiobjective optimization framework is proposed based on the FEM and an adaptive local search strategy. A hybrid multiobjective genetic algorithm is then incorporated into the proposed framework to solve the problem at hand. Extensive experiments carried out confirm that our algorithm outperforms five other well-known multiobjective algorithms in solving the problem.