Genetic algorithms have been used extensively in solving complex solution-space search problems. However, certain problems can include multiple sub-problems in which multiple searches through distinct solution-spaces are required before the final solution combining all the sub-solutions is found. This paper presents a generic design of genetic algorithms which can be used for solving complex solution-space search problems that involve multiple sub-solutions. Such problems are very difficult to solve using basic genetic algorithm designs that utilize a single gene-set per chromosome. The suggested algorithm presents a generic solution which utilizes both multi-gene-set chromosomes, and an adaptive gene mutation rate scheme. The results presented from experiments done using an automatic graphical user interface generation case study, show that the suggested algorithm is capable of producing successful solutions where the common single-gene-set design fails.