Journal of Intelligent Manufacturing, 15, 187±199, 2004 # 2004 Kluwer Academic Publishers. Manufactured in The Netherlands. Employing subgroup evolution for irregular-shape nesting AMBER D. FISCHERandC. H. DAGLI Smart Engineering Systems Laboratory, Department of Engineering Management, University of Missouri-Rolla, Rolla, MO 65401, USA Received February 2002 and accepted June 2003 This paper introduces a new method to solve the irregular-shape, full-rotation nesting problem by a genetic algorithm. Layout patterns are evolved in hierarchical subgroups to facilitate the search for an optimal solution in such a complex solution space. The genotype used in the genetic algorithm contains both the sequence and rotation for each shape, requiring new genetic operators to manipulate a multi-type genetic representation. A lower-left placement heuristic coupled with matrix encoding of the shapes and plate prevents overlap and constrains the solution space to valid solutions. This new method is able to ef®ciently search the solution space for large problems involving complex shapes with 360 degrees of freedom. The algorithm generates better solutions than previously published evolutionary methods. Keywords: Nesting, genetic algorithms, optimization, stock-cutting, geometric modeling, evolutionary computation 1. Introduction to the nesting problem Over the last 40 years, a considerable amount of research from numerous areas, such as engineering, management science, computer science, and opera- tions research, has been invested in developing algorithms to produce solutions for operations requiring pieces to be cut from stock material. Decreasing the waste of material by a single percentage can mean millions in cost savings for many industries (Dagli and Tatoglu, 1987; Qu and Sanders, 1987; Li and Milenkovic, 1995). This optimization problem is generally referred to as the nesting problem. The problem has special signi®cance in those such as the aerospace industry where there is a growing use of composites associated with high costs and long preorder times (Cheng et al., 1994). The two-dimensional nesting problem involves optimizing the layout of multiple pieces (geometric shapes) onto a predetermined area, a plate, without allowing overlap of the pieces. The uncovered surface area on the plate, resulting from the spaces between and around the placed pieces, is wasted material. The objective of the nesting function is to minimize this wasted material by constructing an optimal layout of the given pieces. Figure 1 shows a simple example of the two-dimensional nesting problem. The general nesting problem has been solved by a variety of search algorithms utilizing many different model representations, for a review see Cheng et al. (1994), or Dowsland and Dowsland (1995). Most search mechanisms, however, obtain optimal results only in smaller simpler problems, usually nesting only a few rectangular shaped pieces with 90  or no rotation. As the problem becomes larger and shapes that are more complex are nested, the solution space grows exponentially. A more powerful search algo- rithm coupled with an ef®cient model representation is necessary for obtaining optimal nested layouts in the complex nesting problem. One such powerful search mechanism is the genetic algorithm. In genetic algorithms, the initialization of the population and the creation of the ®tness function de®ne the problem to be solved and the landscape of the problem space. The rest of the algorithm is a simulation of the evolution process, where each