Soft Comput DOI 10.1007/s00500-017-2650-3 FOUNDATIONS Mathematical analysis of schema survival for genetic algorithms having dual mutation Apoorva Mishra 1 · Anupam Shukla 1 © Springer-Verlag Berlin Heidelberg 2017 Abstract Genetic algorithms are widely used in the field of optimization. Schema theory forms the foundational basis for the success of genetic algorithms. Traditional genetic algo- rithms involve only a single mutation phase per iteration of the algorithm. In this paper, a novel concept of genetic algo- rithms involving two mutation steps per iteration is proposed. The purpose of adding a second mutation phase is to improve the explorative power of the genetic algorithms. All the pos- sible cases regarding the working of the proposed variant of the genetic algorithms are explored. After a meticulous anal- ysis of all these cases, three lemmas are proposed regarding the survival of a schema after the application of the dual mutation. Based on these three lemmas, a theorem is proved, and a mathematical expression representing the probability of survival of a schema after the application of the crossover and dual mutation is derived. This expression provides a new insight about the penetration of a schema for such scenario and improves our understanding of the functioning of this modified form of the genetic algorithm. Keywords Genetic algorithms · Crossover · Dual mutation · Schema · Schema survival Communicated by A. Di Nola. B Apoorva Mishra apoorvamish1989@gmail.com Anupam Shukla dranupamiiitm@gmail.com 1 Soft Computing and Expert System Laboratory, ABV -Indian Institute of Information Technology and Management, Gwalior, Madhya Pradesh 474010, India 1 Introduction Genetic algorithms mimic the functioning of natural evolu- tion and try to use its power to solve several optimization problems. There are many areas where they have been suc- cessfully used (Basagoiti and Rodriguez 2016; Hsu and Cho 2015; Li et al. 2014; Mehboob et al. 2016; Nogueira et al. 2016; Shih et al. 2016; Thi et al. 2016). Some of the basic operators used in genetic algorithms are selection, crossover and mutation (Shukla et al. 2010). As per the basic principle of the genetic algorithm, these operators are repeat- edly applied one after the other until the desired solution is obtained. Various variants of these basic operators have also been proposed (Banerjee 2013; Faraji and Naji 2014; Mishra and Shukla 2016; Qiongbing and Lixin 2016; Uzor et al. 2016) to improve the performance of the traditional genetic algorithm and have been successfully applied to solve differ- ent optimization problems. 1.1 Working of the traditional genetic algorithm The working of the traditional genetic algorithms is illus- trated in Fig. 1. The algorithm begins by initializing a random set of the population, and then, the fitness of each individual as well as that of the entire population as a whole is eval- uated. Then, it is checked that whether the solution set meets the termination criteria? If the termination criteria are satisfied, then the algorithm is stopped, else, the follow- ing genetic operators are applied in succession: selection, crossover and mutation, to obtain the next-generation popu- lation, and then, again the fitness in evaluated and the process repeats. 123