Integrated Mapping and Scheduling (IMS) problems can be found in many domains, such as electronic design automation (EDA) and modern manufacturing systems. Optimization algorithms to solve the IMS problems can be used to minimize execution time, implementation cost, energy consumption, etc. Genetic Algorithms (GAs) are powerful evolutionary algorithms for tackling many of such IMS optimization problems. By utilizing biological principles like selection, crossover, and mutation, GAs excel in generating high-quality solutions. Chromosome encoding and decoding, in addition to evolutionary operators, significantly influence GA's efficiency. This paper introduces a relative-priority genetic algorithm (RPGA), a novel GA for IMS problems, such as those in EDA. RPGA employs a unique encoding scheme tailored for IMS problems, especially those with OR nodes representing alternative operation paths. It encodes the relative priority of an operation in a chromosome, which can be divided into two parts: one for path selections and the other for operation scheduling and mappings. Efficient decoding of every chromosome into a solution is facilitated through the concept of a ready operation set. The study extensively compares RPGA and established meta-heuristics using a benchmark set. The experimental results demonstrate that RPGA achieves high solution quality and rapid convergence.