Author:  Yung, Chiho 
Title:  Heuristic search for nonguillotine cutting stock problem by simulated annealing and genetic algorithm 
Degree:  M.Sc. 
Year:  1998 
Subject:  Cutting stock problem  Mathematical models Heuristic programming Mathematical optimization Genetic algorithms Simulated annealing (Mathematics) Hong Kong Polytechnic University  Dissertations 
Department:  Multidisciplinary Studies Dept. of Applied Mathematics 
Pages:  82, [112] leaves : ill. ; 30 cm. + computer disk 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b1446724 
URI:  http://theses.lib.polyu.edu.hk/handle/200/558 
Abstract:  Heuristic Search for Nonguillotine Cutting Stock Problem by Simulated Annealing and Genetic Algorithm The main idea of this dissertation comes from two papers. In 1996, S. Jakobs published a paper about genetic algorithm (GA) on twodimensional bin packing problem. After a year, K. K. Lai had applied the simulated annealing algorithm (SA) on nonguillotine cutting stock problem (CSP) in his paper. They used the same model (permutation) to formulate this problem, with different decoding processes which is an important element in GA and SA. One of the objectives of this dissertation is to make comparisons of these evolutionary algorithms or heuristic search methods (i.e. SA and GA) and the decoding processes. For simplicity, nonguillotine CSP is a topic of finding a packing pattern of lowest wastes. In the past, this problem has been formulated as linear programming. Since no deterministic polynomialtime algorithm has been found for this problem, it is a NPhard problem. So heuristic methods have been devised to tackle CSP but they often involve complicated procedures and require hard knowledge. In recent year, the heuristic search methods (i.e. SA, GA and TA etc) are very successful in applying on many combinatorial problems, e.g. scheduling, graph coloring and travelling salesman problem. They are easy to implement and does not require much mathematical analysis on the specific problem. In nonguillotine CSP, a permutation (string) represents the sequence in which the rectangular pieces are packed into a plate. The magnitude of search space for this representation is n! where n = number of pieces for packing. Two decoding processes (decoders) are studied in this dissertation; they are bottom left (BL) and difference process (DP). Actually each one is a set of rules to pack pieces according to the sequence in the permutation. In other words, a permutation can be transformed into the corresponding packing pattern by these decoders. Then the fitness value (i.e. waste or trim loss) of the permutation can be calculated at once. If bottom left is chosen as the decoder, pieces will enter plate by alternative downwards and leftwards movements. But if difference process is chosen, pieces will be packed into the space interval which is nearest to leftbottom corner of the plate. After defining the permutation model and decoders, the algorithms SA and GA are used to search a suboptimal solution (i.e. the string with low trim loss). Since they are heuristic search methods, it does not guarantee that the optimum must be found. Unlike the random search, these heuristic methods have their own mechanisms and search the solution based on some criteria. SA differs from the traditional iterative improvement algorithm, it has a chance of accepting worse solutions in each iteration. The purpose is to escape from the current local optimum and drive into a more promising solution. On the other hand, GA is an iterative algorithm that maintain a pool of solutions (population) at each iteration. New pool of solutions is formed from the old one by genetic operations: crossover and mutation. Eight test problems from small to medium sizes are used in the experiment. Both SA and GA are tested by changing their parameters. There are several new findings and conclusions from the results, which are not discussed in details in previous papers. SA will gives us a better solution in CSP when the temperature is not raised too high at the start and the cooling scheme has gentle rate of decreasing. The mutation rate of GA in this dissertation should be large and near to 1.0 because it converges to suboptimum faster than the classical approach. In crossover operators, GA prefers SJX and OBX to PMX, OX and CX from their performances. For heuristic search methods comparison, GA is superior to SA in both performance and efficiency. For decoders comparison, difference process is better than bottom left algorithm but the later decoder requires less time than the former decoder in a decoding process. 
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