Vol. 33, No. 9 ACTA AUTOMATICA SINICA September, 2007 An Improved Heuristic Recursive Strategy Based on Genetic Algorithm for the Strip Rectangular Packing Problem ZHANG De-Fu 1 CHEN Sheng-Da 1 LIU Yan-Juan 1 Abstract An improved heuristic recursive strategy combining with genetic algorithm is presented in this paper. Firstly, this method searches some rectangles, which have the same length or width, to form some layers without waste space, then it uses the heuristic recursive strategies to calculate the height of the remaining packing order and uses the evolutionary capability of genetic algorithm to reduce the height. The computational results on several classes of benchmark problems have shown that the presented algorithm can compete with known evolutionary heuristics. It performs better especially for large test problems. Key words Strip packing problems, heuristic, recursive, genetic algorithm 1 Introduction Many industrial applications, which belong to cutting and packing problems, have been found. Each application incorporates different constraints and objectives. For ex- ample, in wood or glass industries, rectangular components have to be cut from large sheets of material. In warehousing contexts, goods have to be placed on shelves. In newspapers paging, articles and advertisements have to be arranged in pages. In the shipping industry, a batch of objects of var- ious sizes has to be shipped to the maximum extent in a larger container, and a bunch of optical fibers has to be accommodated in a pipe with perimeter as small as possi- ble. In very-large scale integration (VLSI) floor planning, VLSI has to be laid out. These applications have a similar logical structure, which can be modeled by a set of pieces that must be arranged on a predefined stock sheet so that the pieces do not overlap with one another, so they can be formalized as the packing problem [1] . For more extensive and detailed descriptions of packing problems, the reader can refer to [1∼3]. A two-dimensional strip rectangular packing problem is considered in this paper. It has the following characteris- tics: a set of rectangular pieces and a larger rectangle with a fixed width and infinite length, designated as the con- tainer. The objective is to find a feasible layout of all the pieces in the container that minimizes the required con- tainer length and, where necessary, takes additional con- straints into account. This problem belongs to a subset of classical cutting and packing problems and has been shown to be non-deterministic polynomial (NP) hard [4, 5] . Opti- mal algorithms for orthogonal two-dimension cutting were proposed in [6, 7]. Gilmore and Gomory [8] solved prob- lem instances to optimality by linear programming tech- niques in 1961. Christofides and Whitlock [9] solved the two-dimensional guillotine stock cutting problem to opti- mality by a tree-search method in 1977. Cung et al. [10] developed a new version of the algorithm proposed in Hifi and Zissimopolous that used a best-first branch-and-bound approach to solve exactly some variants of two-dimensional stock-cutting problems in 2000. However, these algorithms might not be practical for large problems. In order to solve large problems, some heuristic algorithms were developed. Received June 20, 2006; in revised form October 24, 2006 Supported by Academician Start-up Fund (X01109), 985 Informa- tion Technology Fund (0000-X07204) in Xiamen University 1. Department of Computer Science, Xiamen University, Xiamen 361005, P. R. China DOI: 10.1360/aas-007-0911 The most documented heuristics are the bottom-left (BL), bottom-left-fill (BLF) methods, and other heuristics [11∼13] . Although their computational speed is very fast, the so- lution quality is not desirable. Recently, genetic algo- rithms and its improved algorithms for the orthogonal pack- ing problem were proposed because of their powerful op- timization capability [14∼17] . Kroger [14] used genetic algo- rithm for the guillotine variant of bin packing in 1995. Jakobs [15] used a genetic algorithm for the packing of poly- gons using rectangular enclosures and a Bottom-left heuris- tic in 1996. Liu et al. [16] further improved it. Hop- per and Turton [18] evaluated the use of the BLF heuris- tic with genetic algorithms on the nonguillotine rectangle- nesting problem in 1999. In addition, an empirical in- vestigation of meta-heuristic and heuristic algorithms of the strip rectangular packing problems was given by [19]. Recently, some new models and algorithms were devel- oped by [20∼26]. For example, quasi-human heuristic [20] , constructive approach [21, 22] , a new placement heuristic [23] , heuristic recursion (HR) algorithm [24] , and hybrid heuristic algorithms [25, 26] were developed. These heuristics are fast and effective, especially, the best fit in [23] not only is very fast, but also finds better solutions than some well-known metaheuristics. In this paper, an improved heuristic recursive algorithm that combines with genetic algorithm is presented to solve the orthogonal strip rectangular packing problem. The computational results on a class of benchmark problems show that this algorithm can compete with known evolu- tionary heuristics, especially in large test problems. The rest of this paper is organized as follows. In Section 2, a clear mathematical formulation for the strip rectangu- lar packing problem is given. In Section 3, the heuristic recursive algorithm is presented, and an improved heuris- tic recursive algorithm (IHR) is developed in detail. In Section 4, the GA+IHR algorithm is proposed. Computa- tional results are described in Section 5. Conclusions are summarized in Section 6. 2 Mathematical formulation of the problem Given a rectangular board of given width and a set of rectangles with arbitrary sizes, the strip packing problem of rectangles is to pack each rectangle on the board so that no two rectangles overlap and the used board height is min- imized. This problem can also be stated as follows.