Abstract
This chapter introduces biased random-key genetic programming, a new metaheuristic for evolving programs. Each solution program is encoded as a vector of random keys, where a random key is a real number randomly generated in the continuous interval [0,1]. A decoder maps each vector of random keys to a solution program and assigns it a measure of quality. A Program-Expression is encoded in the chromosome using a head-tail representation which is later transformed into a syntax tree using a prefix notation rule. The artificial simulated evolution of the programs is accomplished with a biased random-key genetic algorithm. Examples of the application of this approach to symbolic regression are presented.

Abstract
This paper addresses a two-dimensional (2D) non-guillotine cutting problem, where a set of small rectangular items of given types has to be cut from a large rectangular stock plate having defective regions so as to maximize the total value of the rectangles cut. The number of small items of each item type which can be cut from the large object is unrestricted. A novel MIP model and a hybrid approach combining a novel placement procedure with a biased random-key genetic algorithm (BRKGA) are presented. The parameters used by the novel placement procedure for the development of a cutting plan are evolved by the BRKGA. The management of the free spaces and of the defects uses a maximal-space representation. The approach is evaluated and compared to other approaches by means of a series of detailed numerical experiments using 5414 benchmark instances taken from the literature. The experimental results validate the quality of the solutions and the effectiveness of the proposed algorithm.

Abstract
This paper presents a hybrid genetic algorithm for the simple assembly line problem, SALBP-1. The chromosome representation of the problem is based on random keys. The assignment of the operations to the workstations is based on a heuristic priority rule in which the priorities of the operations are defined by the chromosomes. A local search is used to improve the solution. The approach is tested on a set of problems taken from the literature and compared with other approaches. The computation results validate the effectiveness of the algorithm.

Abstract
This paper presents a biased random-key genetic algorithm for the resource constrained project scheduling problem. The chromosome representation of the problem is based on random keys. Active schedules are constructed using a priority-rule heuristic in which the priorities of the activities are defined by the genetic algorithm. A forward-backward improvement procedure is applied to all solutions. The chromosomes supplied by the genetic algorithm are adjusted to reflect the solutions obtained by the improvement procedure. The heuristic is tested on a set of standard problems taken from the literature and compared with other approaches. The computational results validate the effectiveness of the proposed algorithm.

Abstract
Random-key genetic algorithms were introduced by Bean (ORSA J. Comput. 6:154-160, 1994) for solving sequencing problems in combinatorial optimization. Since then, they have been extended to handle a wide class of combinatorial optimization problems. This paper presents a tutorial on the implementation and use of biased random-key genetic algorithms for solving combinatorial optimization problems. Biased random-key genetic algorithms are a variant of random-key genetic algorithms, where one of the parents used for mating is biased to be of higher fitness than the other parent. After introducing the basics of biased random-key genetic algorithms, the paper discusses in some detail implementation issues, illustrating the ease in which sequential and parallel heuristics based on biased random-key genetic algorithms can be developed. A survey of applications that have recently appeared in the literature is also given.

Abstract
This paper addresses a constrained two-dimensional (2D), non-guillotine restricted, packing problem, where a fixed set of small rectangles has to be placed into a larger stock rectangle so as to maximize the value of the rectangles packed. The algorithm we propose hybridizes a novel placement procedure with a genetic algorithm based on random keys. We propose also a new fitness function to drive the optimization. The approach is tested on a set of instances taken from the literature and compared with other approaches. The experimental results validate the quality of the solutions and the effectiveness of the proposed algorithm.