AN EVOLUTIONARY APPROACH TO OPTIMIZATION OF A LAYOUT CHART

Transcription

1 13 AN EVOLUTIONARY APPROACH TO OPTIMIZATION OF A LAYOUT CHART Eva Vona University of Ostrava, 30th dubna st. 22, Ostrava, Czech Repubic e-mai: Eva.Vona@osu.cz Abstract: This artice presents the use of a genetic agorithm to find an optima ayout for the pacement of reguar patterns of fixed sizes and simpe shapes to minimize the waste. Experiments on various pattern designs indicate that genetic agorithms can effectivey be used to obtain highy efficient soutions. Key words: Genetic agorithms, shape ayout optimization. 1. Introduction The pacement of pieces to use the minima amount of surface is important for industry. Each piece presents a chaenge for finding a good pacement soution. Much effort has been devoted to automate this process by using artificia inteigence and optimization techniques. This paper presents a methodoogy to generate a suitabe shape ayout soution using a genetic agorithm [1]. Genetic agorithms seem to be we suited for shape pacement, especiay since it does not suffer from oca minima probems. Another potentia approach coud be based on symboic methods preferaby automated reasoning in fuzzy ogic [7]. The variations in various tasks on the use of genetic agorithms stems from the representation and the use of differing crossover and mutation methods. In the next sections the probem of representation, agorithm, and experimenta resuts wi be presented. Figure 1 shows an exampe of using convex shapes in the experiment, where these shapes are modeed as poygona objects. Each of the shape has a defaut initia orientation, but may be rotated. A soution or individua is a structure with the foowing format: S = [(P 1, O 1 ), (P 2, O 2 ),..., (P n, O n ), L] where S is the soution, P represents each piece, O is the orientation of the piece: 0 for 0 degrees, or 1 for 90 degrees, 2 for 180 degrees or 3 for 270 degrees, and L is the ength (cost) of the soution. Figure 1: Type of pieces. 2. Shape ayout soution using a genetic agorithm Particuar individua are strips with the fixed width W (e.g. W = 2, where is a constant). Every individua I k = {(P 1, O 1 ), (P 2, O 2 ),..., (P 10, O 10 )}, O i {0, π/2, π, 3π/2} is represented by its chromosome that is a define set of pieces (see Figure 1) with their own orientations. We experimented with ten pieces (e.g. two rectange pieces, four square pieces, and four triange pieces). Each of these ten pieces is used just once in the chromosome. This mode of chromosome is simiar to that in [2]. Each piece is paced starting at the upper-eft edge of the strip. If there is no space to pace the piece, we move downwards unti there is a

2 14 space or we run to the right on the strip. When pacing a new piece we must check that a space is avaiabe. We do this by means of a simpe 2-D graphics agorithm for checking that none of the vertices of our poygon is inside another, previousy paced poygon [3]. To encode an individua into a string, we use a triange grid (see Figure 2) Figure 2: A triange grid The purpose of this paper fits in the optimization of appare shape ayout. The genetic agorithm used in this paper is outined beow. With its assistance the best optima popuation is made from the set of individuas. The initia popuation is created by randomy generating N individuas. Number of individuas in the popuation was constant during the whoe cacuation. The fitness function vaue of each chromosome is defined as a reciproca vaue of its strip s ength: 1 F = = = (1) L k k Then, for each fitness function the probabiity of reproduction of its existing individua is cacuated by means of standard method (see [4]). A of the cacuated fitness function vaues of the two consecutive generations are sorted descending and individuas attached to the first haf creates the new generation. A new individua may be created by either a crossover or a mutation. The crossover runs in two foowing steps: we pick a suitabe chromosome from our popuation to crossover at random. After seecting a chromosome to become a part of a new individua, the pieces in that chromosome wi be interchanged. We generate a number (a strip s position) that is bounded above by its strip s ength. The first (second) new individua incudes the first (second) substring of the parent and then we insert a the remaining pieces from this parent in its second (first) substring - that are randomy ocated, but their orientations are given - to compete our new individua. If the input condition of mutation is fufied (e.g. if a randomy number is generated that is equa to the defined constant), one of the individuas is randomy chosen and its genetic representation is randomy chosen too. The mutation can run in this form: We choose one pace of the individua randomy and then we exchange a piece s orientation in this position. The finding of optima popuation is finished when the popuation achieves the maxima generation or the best soution cannot be further improved in its fitness function vaue (e.g. in the ength of its strip).

3 Acta Fac. Paed. Univ. Tyrnaviensis, Ser. C, 2005, no. 9, pp Experimenta resuts Once the system has converged, we pick the individua with the best fitness and report its configuration as the soution. We experimented with a popuation that contained 30 individua and our popuation size throughout our experiments was constant. Every individua consists of a ten pieces from the defined set of pieces (e.g. two rectange pieces, four square pieces, and four triange pieces), see Figure 1. The initia popuation was created by randomy generating individuas. Each of pieces was paced in the chromosome (e.g. a strips with the fixed width W = 2, where is a constant) at random and its orientation was generated from the set of the possibe orientations. The best and the worst individuas from the initia popuation are shown in Figure 3. (a) W=2 L=6 (b) W=2 L=9 Figure 3: The best (a) and the worst (b) individua from the initia popuation. Our experimenta resuts indicate that our genetic agorithm is reasonaby good. The cacuation is finished in the 569th generation to be characterised by the popuation of the same individua. The figure 4 iustrates the best individua (e.g. soution) of the fina 1 popuation: its fitness function vaue is the foowing F = = = = 0.2. If we compare L 5 5 our resuts to those of human experts, we can observe that it is the ony possibe soution. W=2 L=5 Figure 4: The best individua from the fina popuation.

4 16 In Figure 5 the history of fitness function vaues is shown as: (a) the best individua in the popuation and (b) the average individua in the popuation during the whoe cacuation. Other numerica simuations give simiar resuts. Fitness function is represented here in a reative way so that vaue one means the upper-most possibe fitness function vaue and vaue zero means the owest fitness function vaue. The three parameters in this method that must be defined for each probem are: the size of the popuation (e.g. 30 in our experiment), the probabiity of crossover (e.g. 0.5 in our experiment), and the probabiity of mutation (e.g in our experiment). FITNESS 1,0 Fitness average Fitness max 0,5 0, number of generation Figure 5: The history of the media and the best fitness function vaues during cacuation. A comparison of our resuts with other researchers [5, 6] is rather difficut, since the overa efficiency depends on the shape of the patterns used. We need to compare our technique with that of other researchers using the same test data. Before we do that a standard test data set has to be estabished. 4. Concusion The resuts from experiments on various pattern designs indicate that genetic agorithms can effectivey be used to obtain highy efficient soutions. References [1] GOLDBERG, D. E. Genetic Agorithms in Search, Optimization, and Machine Learning, Reading MA, Addison-Wesey, pp. 1-23, [2] PARGAS, R. P. and JAIN, R. A Parae Stochastic Optimization Agorithm for Soving 2D Bin Packing Probems, IEEE Proc. of the 9th Int. Conf. on Artificia Inteigence for Appications, pp , [3] HARRINGTON, S. Computer Graphics, New York, McGraw-Hi, pp. 3-5, 70-76, [4] LAWRENCE, D. Handbook of genetic agorithms, Van Nostrand Reinhod, New York [5] BOUNSAYTHIP, C., MAOUCHE, S. and NEUS, M. Evoutionary Search Techniques Appication to Automated Lay-Panning Optimization Probem, IEEE Internationa Conference on Inteigent Systems for the 21st Century, v. 5, pp , 1995.

5 17 [6] MADARASMI, S. and SIRIVAROTHAKUL, P. Layout of Garment Patterns for Efficient Fabric Consumption,Internationa Technica Conference on Circuits/Systems, Computers and Communications (ITC-CSCC 2002), Phuket, Thaiand, pp [7] HABIBALLA, H. Non-causa Resoution in Fuzzy Predicate Logic with Evauated Syntax (background and impementation). In Proceedings of Internationa conf. The Logic of Soft Computing IV, Ostrava, 2005.