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In this paper, a new genetic algorithm (GA) is presented for solving the multi-mode resource-constrained project scheduling problem (MRCPSP) with minimization of project makespan as the objective subject to resource and precedence constraints. A random key and the related mode list (ML) representation scheme are used as encoding schemes and the multi-mode serial schedule generation scheme (MSSGS) is considered as the decoding procedure. In this paper, a simple, efficient fitness function is proposed which has better performance compared to the other fitness functions in the literature. Defining a new mutation operator for ML is the other contribution of the current study. Comparing the results of the proposed GA with other approaches using the well-known benchmark sets in PSPLIB validates the effectiveness of the proposed algorithm to solve the MRCPSP.