Applied Intelligence 23, 9–20, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.
Genetic Algorithm Coding Methods for Leather Nesting ALAN CRISPIN, PAUL CLAY AND GAYNOR TAYLOR School of Technology, Leeds Metropolitan University, Calverley Street, Leeds LS1 3HE, England
[email protected] [email protected]
TOM BAYES SATRA Technology Centre, Rockingham Road, Kettering, Northamptonshire, N16 9JH, England
[email protected]
DAVID REEDMAN R&T Mechatronics Ltd. The Cottage, Main Street, Wartnaby, Melton Mowbray, Leicestershire, LE14 3HY, England
[email protected]
Abstract. The problem of placing a number of specific shapes in order to minimise waste is commonly encountered in the sheet metal, clothing and shoe-making industries. The paper presents genetic algorithm coding methodologies for the leather nesting problem which involves cutting shoe upper components from hides so as to maximise material utilisation. Algorithmic methods for computer-aided nesting can be either packing or connectivity driven. The paper discusses approaches to how both types of method can be realised using a local placement strategy whereby one shape at a time is placed on the surface. In each case the underlying coding method is based on the use of the no-fit polygon (NFP) that allows the genetic algorithm to evolve non-overlapping configurations. The packing approach requires that a local space utilisation measure is developed. The connectivity approach is based on an adaptive graph method. Coding techniques for dealing with some of the more intractable aspects of the leather nesting problem such as directionality constraints and surface grading quality constraints are also discussed. The benefits and drawbacks of the two approaches are presented. Keywords: computer-aided nesting, genetic algorithms, encoding, leather, image processing, packing, connectivity, optimisation
1.
Introduction
The general two-dimensional cutting stock problem can be defined as follows. Given a set of N shapes find the optimal non-overlapping configuration of all shapes on a sheet. The N shapes can consist of a smaller set of shapes that are repeated a number of times to provide consistent parts for a number of assembled items. The
cutting stock (or trim loss) problem generally refers to the case where the surface and parts are rectangular. However, in practical problems as found in the textile and leather shoe making industries the parts are irregularly shaped and in the case of shoe manufacture the surface shape is irregular being a hide. Consequently, the leather nesting problem involves placing a set of plane irregularly shaped parts on a plane irregularly
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shaped surface which is also characterised by further constraint requirements (see Section 2). Many packing-driven solutions to the cutting stock problem have been reported in the literature [1]. Anand et al. [2] and Bounsaythip et al. [3] have explored genetic algorithm approaches. One of the main difficulties reported was finding an appropriate encoding strategy for mapping shapes evolved by the genetic algorithm to non-overlapping configurations. Anand [2] used an unconstrained approach whereby the genetic algorithm was allowed to evolve overlapping configurations and post processing was subsequently used to check and correct for shape intersection. Bounsaythip et al. [3] have used an order based tree encoding method to pack components in strips. Kahng [4] discusses the preoccupation with packingdriven as opposed to connectivity-driven problem formulations in the context of floor planning (i.e. automated cell placement for VLSI circuits) and suggests that approaches based on a connectivity centric should be used. This paper explores genetic algorithm based coding methods for both a packing driven and connectivity driven solution to the leather-nesting problem. Section 2 describes the leather-nesting problem. Section 3 describes shape placement using the no-fit polygon. Section 4 develops a packing driven genetic algorithm solution based on developing a local space utilisation measure. In Section 5, a connectivity driven genetic algorithm solution is presented based on creating a graph at time of placement such that rules are used to prevent invalid positioning. Section 6 discusses results and the benefits and drawbacks of the two approaches. Finally conclusions are drawn.
2.
Problem Description
The shoe manufacturing process starts with the cutting of shoe upper components from a hide with parts nested together as far as possible to maximise material usage. A significant proportion of the manufacturing cost of a pair of shoes is invested in the natural raw material and so the efficient utilisation of this resource is of prime importance. Small improvements in yield can have a dramatic effect on profitability. In leather nesting constraints are placed on the selection process by external practicalities. In particular certain areas of the hide may be unsuitable for a given class of components. Such areas may be identi-
fied as unsuitable because of the variation in hide quality with position e.g. defects such as scar tissue and holes that may occur randomly. In order to make decisions about placement the skilled cutter (and any proposed automation) requires implicit knowledge of the subsequent processes. During the lasting or 3D shaping of the shoe upper certain components undergo large elastic and visco-elastic distortions. The presence of singular defects in such components may lead to a mechanical failure during lasting and will certainly be unacceptable cosmetically in the finished product. There is also a need to understand the anisotropy of the elastic properties of leather and their effect on the shaping process. This is known as identifying the lines of tightness, which are generally held to follow the skeletal structure. Conventionally shoe components are cut “tight to toe” so that, when assembled, the stiffer elastic axis runs consistently along the shoe. Any proposed computer-aided nesting system must take into account the constraint criteria outlined above. In addition any cosmetic defects present in the leather must either be discarded or must appear in areas of the shoe which are hidden from casual view. The hide data files used for this research contain point co-ordinate information on the edge outline, holes and grade regions (quality zones) from real hides scanned at a shoe factory. Region grading conforms to a grading system commonly used in shoemaking, which ranges from grade 1 regions where material quality is rated between 95–100% down in steps of 5% to grade 5 regions where material quality is rated between 75–80%. An image-point dual was used in the work allowing images of shapes to be created from part co-ordinates and vice versa [5]. Figure 1 shows a typical hide image with grade regions marked. To constrain the placement of shapes that require high quality leather a threshold can be applied to the grade image on a level determined by the lowest grade that a shape may be placed within. Any lower grades are removed from the hide. This approach completely restricts a shape from entering a poorer grade area than required. This work has used a tightness vector map, which generically follows the skeletal structure (see Fig. 1). This allows an error difference angle to be calculated between the tightness vector and shape direction vector. A shape is placed so its direction is constrained to a maximum deviation value either side of the tightness vector.
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Figure 1. Hide image with directionality and grade constraints shown. Darker regions represent poorer quality grade material. Tightness vector arrows indicate the required placement direction.
3.
Shape Placement Using NFP
Optimal nesting can only be achieved when shapes are touching since any small gaps between two shapes are usually unusable and so are wasted. The no-fit polygon (NFP) [6, 7] is the path traced by any selected reference point on one shape as it circumnavigates around another shape. Selection of the reference point will translate the NFP relative to the other shape. In Fig. 2 the reference point has been taken as the shape centroid. The NFP can be used to ensure that each shape placed touches the hide edge or another shape without overlap. A number of different methods have been devised to calculate the NFP between two shapes involving the addition of ordered vectors describing the polygonal
edges of the individual shapes [8]. The basic algorithm only works correctly for convex polygons and a number of adjustments have been made to accommodate the relationship between both simple non-convex [9] and more complicated non-convex shapes [10]. Our approach to calculating the NFP is based on image processing techniques [5, 11]. The NFP is found from the Minkowski sum, which is the convolution between two shape images at a set orientation. The calculated Minkowski sum for two leather components used in shoe making is shown in Fig. 2. The NFP is the boundary between the area covered by the Minkowski sum and those areas not covered by the sum. Four-connected shape boundary images are used to calculate the Minkowski sum with the NFP extracted using morphological operators. As the angle
Figure 2. Minkowski sum found by convolution between two shape images at a set orientation. The NFP is the boundary and is dependent on the angle between the two shapes as shown in the second diagram.
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Figure 3.
An example of a shape-hide no-fit polygon (NFP) and window region centred on point labelled A.
between shapes changes the NFP jumps from one state to another (i.e. there is a non-linear discontinuous change). 4.
Packing Driven Solution
A packing driven solution requires that shapes are placed such that space utilisation is maximised. The method developed for this work is based on a local placement strategy where shape parts are placed on the hide one at a time in sequence. Once a shape is placed it is not changed becoming part of a dead region in which it is not possible to place further parts. The local placement approach requires using a local measure for space utilisation such that the sum approximates to the final space utilisation measure. The alternative approach is to use a global placement method in which whole lays are repeatedly created on a hide and tested directly against a global space utilisation measure. Heistermann and Lengauger [12] discuss the advantages of using a local placement strategy over that of a global placement strategy in terms of nesting time requirements. The strategy of our packing algorithm is based on placing a shape at a point on the shape/hide NFP (or a shape/shape NFP) to maximise local space utilisation. Calculations are performed within a window region. The window is initially positioned around the point at the rear of the spine of the hide nearest to the image corner. In Fig. 3 this point is labelled A and the first shape is placed in the window region shown such that the shape which fits closest to the edge is chosen. The window is moved (i.e. stepped) so that it is centred on
the previously placed shape and the process of finding the best local placement solution within the new window region is repeated. Our window placement strategy imitates the placement strategies of a professional lay-planner and promotes clustering and runs of similar shapes. The NFP calculations are only performed within the local window region, thereby reducing the number of computational operations required as compared to those that would be required if the whole hide image is used. Figure 4 shows an example of shape placement against a hide edge within a window area. The window is centred on a current point labelled A. The average directionality angle is calculated using the tightness vector map in a box positioned at the centre point of
Figure 4. An example of shape placement within a window region showing the relationship between shape angle θ and lay angle ϕ.
Genetic Algorithm Coding Methods for Leather Nesting the line projected from A at lay angle ϕ to the window edge. The box dimensions for determining the direction angle are made equal to the window step distance. A shape angle θ measured from the horizontal axis is chosen such that it is constrained to ±5◦ of the average tightness angle found. The shape is placed at the point on the shape-hide NFP which is closest to the line projected from A at lay angle ϕ. By generating lay angle sample populations a number of placement solutions are created which can each be assigned a fitness value so that the best placement can be determined. The specification requirement that the maximum deviation for the shape angle should be ±5◦ round the tightness angle found in the region of placement is also used in the optimisation process (see Section 4.1). Efficient utilisation of space requires that shapes are located close to one another. An indication of the proximity of a shape to the hide edge or another shape can be calculated for optimisation purposes. Our approach to proximity measurement is based on filtering the binary images of the shape and the hide using a mask of unity-values. A threshold is applied to generate an outer border around both the hide edge and shape (see Fig. 5). It is observed that shapes that fit closest to the edge have the largest overlap area. Consequently, a fitness value can be calculated as the overlap area between the two borders as this provides a measure of proximity to the edge or another shape. The shapes are placed in order of size since it is easier to find suitable positions for the smaller shapes once the available area becomes limited. It is noted that the largest shape, the vamp (toe piece), is also the shape that requires the highest-grade leather and is to be placed with the tightness of the material (direc-
Figure 5. Fitness value calculation. The fitness value is equal to the free space overlap region shown as the darker region.
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tionality) lined up along the length with either orientation acceptable. The number of shoes required is used to determine how many of each shape type should be placed. 4.1.
Genetic Algorithm Packing Optimisation
The key features of a simple genetic algorithm are: – They start with an initial random population of potential solutions. – The population is indexed in terms of fitness using an objective function. – High fitness members are randomly selected and recombined to produce an offspring. – A small proportion of the offspring mutates to extend the search. – Fitness values for the offspring are calculated. – The offspring and their corresponding fitness values replace poorly performing members of the original population. Iteration is used to generate more and more offspring solutions to yield better approximations to the final optimal solution, see Goldberg [13]. For specific coding details see [14, 15]. The genetic algorithm generates a real value population consisting of lay angles and directionality ratios. As previously discussed the shape angle is calculated from the local tightness vector map using a box region centred on the lay angle line. The directionality ratio value is used to allow a constrained variation of shape angle from the local tightness value calculated in the box region. To utilise the mechanics of evolution a selection of the current set of solutions is chosen to become the parents of the next set of offspring. The tournament selection process is used to generate random pairs of indices to the current solutions and their corresponding fitness values compared, the greater of the two being chosen as a parent. Pairs of these successful parents are then combined using linear recombination to create a new population. The offspring is 60% of the size of the population. A mutation operator randomly alters 5% of the genes (lay angle, directionality ratio) in the offspring to extend the search area and so reduce the chances of becoming stuck at sub-optimal solutions. An elitist reinsertion strategy is used. A typical layplan obtained using the packing coding approach is shown in Fig. 6.
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Figure 6. Lay plan of 14 shoes giving 61.6% coverage (with directionality and grading constraints applied).
5.
Connectivity Driven Solution
An alternative coding strategy can be realised based on a connectivity driven approach. Rather than pack shapes to maximise a local space utilisation measure, the shapes can be placed by calculating the intersection points between NFPs. This image processing technique for NFP calculation allows direct Boolean manipulation for fast identification and removal of crossing points of NFPs. The crossing points of two individual NFPs are required to find the points at which a shape can touch two other shapes at the same time. Possible crossing points of the NFPs may lie within the Minkowski sum between the shape to be placed and shapes other than the ones it is required to touch. These Minkowski sums are used as masks to remove these crossing points from the list of possible solutions and so prevent overlapping placements being chosen. Figure 7 shows three shapes with fixed angles and their associated NFPs. In this example, shape1 is placed at a set position and all other shapes connect to it and each other. Shape 2 can be moved around shape1 along NFP 1-2 and one point on this boundary is selected. Valid positions for shape 3 to touch both shapes 1 and 2 are at any crossing points between NFP 1-3 and NFP 23. Therefore the three shapes are connected via the NFP. This connectivity approach can be scaled to any number of shapes by computing NFP intersection points. If a fourth shape is required to touch shapes 1 and 3, possible solutions may lie such that shape 2 is overlapped by shape 4. To prevent overlapping, all points within the area of the Minkowski sum between shapes 2 and 4 are removed using a mask.
Figure 7. No-fit polygons for three shapes. Shape 1 is placed at the origin and Shape 2 is placed on a chosen point on NFP 1-2, shown as a long dashed line. Shape 3 must be placed on a crossing point of NFP 1-3 and NFP 2-3. The crossing points are shown as asterisks.
5.1.
Graph Topologies
This approach can be described using graphs that specify the connectivity between shapes. A graph [16, 17] is a data structure used to model objects and connections between them and is defined by a set of nodes and a set of edges that connect the nodes together. Figure 8 shows a graph of four connected shapes. The properties of the nodes determine shape type and angle. The edges of the graph determine how shapes touch. The graph of Fig. 8 dictates that shapes 1 and 2 touch all other shapes but shapes 3 and 4 touch only shapes 1 and 2 but not each other. Shapes 3 and 4 may be
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Figure 8. A graph to shape placement relationship which shows the way a graph represents the placement of 4 connected shapes.
transposed and the graph is still valid. Consequently, the positioning of shapes can vary for a single combination of graph and node properties. The graph representation describes how shapes connect with each other but not where. The key concept of a graph-based approach is that the edges represent touching shapes. The no-fit polygon can be used for finding placements for a set of shapes that touch one another and allows the graph connectivity concept to be realised. A number of graph topologies can be used to describe the connectivity between shapes. Fixed topology graphs set the connectivity, while the shape type and orientations are adjusted to create a valid placement within the graph constraints. With this type of topology the complete graph may not always be realised with fixed node properties, since the existence of NFP crossing points are dependent on the properties and the placement of those nodes placed previously. Alternatively, a graph can be created at time of placement using rules that prevent invalid shape placement. This is the approach used in this work. The graph edges are varied to fit the lay that develops using the rule that each shape node is connected so that it touches the previous node and an anchor node. An example of the specific case of this adaptable topology is shown in Fig. 9. In this specific case shape 1 is the anchor for shapes 2 to 4, while shape 5 is unable to touch both shapes 4 and 1 without overlapping another shape. Consequently, shape 2 becomes the new anchor for shapes 5 to 8 until shape 9 is unable to touch both shapes 2 and 8. Similarly shape 3 is the anchor for shapes 9 to 11 and shape 4 is the anchor for all other shapes shown. The shapes build up in a ring around a central anchor shape until no more shapes can touch it and the preceding shape without overlapping another shape. The shape placed directly following the anchor node shape now becomes the new anchor node and the over-
Figure 9. An example of a shape connectivity graph for 16 shapes created where a rule is used to check that edges between nodes can realistically exist. Adjusting node properties (shape type, shape angle, angle for next placement) varies the graph and lay.
lapping shape and those after are placed around this new anchor shape until overlapping occurs again. Using this adaptive graph method to create a valid nonoverlapping lay of shapes the properties of the nodes (shape type, shape angle, angle for next placement) can be optimised to improve coverage by applying a genetic algorithm. 5.2.
Genetic Algorithm Connectivity Optimisation
The genetic algorithm has to evolve a combination of shape order, shape angle and NFP intersection point position in order to nest shapes inside the hide area. Vectors can be used to store shape order (represented as a list of N unique numbers) and shape angles (i.e. a list of angles associated with each shape). To select an NFP intersection point requires generating a placement angle. The closest valid crossing point on the NFPs to this angle is selected for placement. The genetic algorithm chromosome is separated into three separate sections: Shape order
set of 20 unique non-repeating integers. Angle of shape 20 modulo 2π real values. Angle for next placement 19 modulo 2π real values. Each chromosome section has its own population, all with the same number of members. The members in all three sections are indexed by a single position in the population so that a single member will consist of three separate sections from the chromosome lists. On placement the radian value at a chosen locus in
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Figure 10. Typical connectivity based lay-plan of 84 shoe parts giving approximately 58% coverage obtained using the adaptive graph connectivity strategy.
the shape angle chromosome section rotates the shape indicated in the corresponding locus in the shape order chromosome section. Each of the angles in the shape angle chromosome section is associated with the shape at the corresponding locus in the shape order chromosome section. When the shapes are reordered, the associated angles are also reordered to the same positions as the corresponding shapes. Consequently, a shape angle is a property of a shape since it is linked to the shape order. The partially matched crossover (PMX) operator as described in Goldberg [13] is used to reorganise the order in which the shapes are placed. PMX combines the sequences contained in the two parent solutions to create two new offspring solutions consisting of elements of both parents while ensuring that no index within the order sequence is repeated. The PMX operator respects the absolute positions of individual indices in the order sequence promoting chromosome structures dependant on positional attributes. The real valued line recombination operator is used for recombining shape angle vectors and placement angle values. Real valued line recombination [18] is a mathematical approach to the combination of real valued solutions. Offspring are created by finding a randomly weighted average of each individual position in the chromosome. The line recombination operator uses a single weighting value for all positions in the chromosome so that solutions are developed on a straight line between each pair of parents. To associate the properties of shape types it was decided to only allow the shape angles from correspond-
ing shapes in the list to recombine. This requires the angles to be reordered so that the corresponding shapes from all chromosomes are in the same order and returning their offspring to their parent’s individual orders after creation. The result would be that the shape angle properties of a shape would only recombine with a like shape irrespective of shape order. The angles are then reordered to follow their respective loci in the shape order chromosome section as described before. The angles for next placement describe a method of positioning a shape relative to the last shape placed. They remain independent of the shape properties. Observation and analysis of the effect of these values suggest that the angles in the first two loci show the greatest importance to the final lay, while the rest have more subtle consequences. The fitness value is calculated as the hide area multiplied by the square of the ratio of the number of shapes placed to the number of shapes required. If all the required shapes are placed the value is equivalent to the absolute area of the hide, any less increases the fitness value significantly. Figure 10 shows a typical result of nesting components using the connectivity coding approach. The graph spirals in from the edge of hide since the coding method attempts to place shapes so that they connect to the previously placed shape and an anchor shape that is initially the hide edge. 6.
Results
An empirical analysis to measure statistically the quality of the final solution (coverage, time) for the
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Figure 11. (a) Change in average percentage coverage for different window sizes (b) change in average time taken for typical lay of whole hide verse window size (1.2 GHz Athlon running Windows 2000).
developed approaches has been undertaken for a range of parameter settings. For a representative hide, the results for the packing approach can be summarised as follows. (i) Changing window size from no window (search whole image) to a size one third of the image height results in a small change in coverage (from 59 to 60%) for a fixed population size (15) and number of generations (25) See Fig. 11(a). (ii) A window size that is a third of the image height takes 50% less time to obtain the same coverage than the case where no window is used for fixed population size (15) and number of generations (25). It was concluded that a window size one third of the image height yielded good coverage in an acceptable computational time. See Fig. 11(b). (iii) For a fixed window size (1/3 of image height) there tends to be a linear increase in average percentage coverage for the range of populations tested (5,15,25,50). See Fig. 12. The algorithm was evaluated with a setting of 25 generations. The increase in coverage is coupled with a linear increase in time. See Fig. 13. When the algorithm was evaluated with a setting of 10 generations the average percentage coverage curve shows a similar trend. (iv) The average percentage of shapes requiring directionality that are within the ±5◦ specification requirement increases with population size. When the algorithm is evaluated with a setting of 10 generations the average percentage of shapes meeting the directionality specification increases from
Figure 12. Average percentage coverage verses population size for packing algorithm with window size equal to 1/3 image height over 25 runs.
47% with a population size of 5 to 65% with a population size of 50. When the algorithm is evaluated with a setting of 25 generations the average directionality percentage increases from 57% with a population size of 5 to 67% with a population size of 50. The connectivity driven approach requires that a shape be placed so that it touches an anchor shape (initially the hide edge) and the shape previously placed. This promotes a spiral lay arrangement. Initial testing was based on the compaction of 20 shapes from 6 different shape types [19]. The results showed that
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shapes first as used in the packing algorithm approach could increase coverage. It should be noted that because shapes are place at NFP crossing points the GA is tuning an initial good placement. The drawback of the approach is the difficulty of developing coding constraints and recombination methods that prevent graphs creating invalid lays when grading and directionality constraints are applied.
7.
Figure 13. Average time taken versus population size for packing algorithm with window size equal to 1/3 image height over 25 runs (1.2 GHz Athlon running Windows 2000).
(i) Modest coverage improvements (of order 10%) could be obtained by the genetic algorithm optimisation of graph node parameters from the initial lay (see Fig.14). (ii) Large sample populations result in good initial lays. Small percentage coverage improvements occur when the graph connectivity approach is applied to leather nesting. It was also found that placing largest
Conclusions
In order to develop a practically effective solution for leather nesting the research has investigated two coding methodologies both based on no-fit polygon calculations. The first approach is based on a local packing paradigm and demonstrates how a window area is used to imitate the actions of a human lay-planner. The method takes into consideration multiple irregular shape types and part-placement using directionality (lines of tightness) and grading constraints. The GA uses a population consisting of lay angles and directionality ratios to optimise part placement within the window region. The second approach is based on a graph connectivity strategy. Graph nodes are considered to have properties shape order, shape angle and placement angle. Graph edges describe connections between shapes. Graphs are created by checking that edges between nodes can realistically exist using rules to connect shapes at NFP crossing points so that non-overlapping placements are always made. A GA is used to evolve node properties (shape order, shape angle and placement angle) for leather nesting. The results of experiments have shown that both methods can be used for leather lay planning. However, it has been found that the packing driven approach provides a simpler coding solution and produces better lay-plan results for the test cases used. Also the implementation of grading and directionality constraints is more readily realised using this approach. Experiments show that the packing method is competitive with human nesters in terms of material useage.
Acknowledgments Figure 14. Compaction area improvement for connectivity algorithm taken over 30 generations as a ratio of area of convex hull to total area of shapes placed. Population sizes of 5 (solid line), 15 (dotted line) and 25 (dashed line) are shown.
The authors gratefully acknowledge the support of this research by the U.K. E.P.S.R.C. grant on “A Genetic Algorithm Approach to Leather Nesting Problems”, reference number: GR/M82110.
Genetic Algorithm Coding Methods for Leather Nesting
References 1. K.A. Dowsland and W.B. Dowsland, “Solution approaches to irregular nesting problems,” European Journal of Operations Research Elsevier, vol. 84, pp. 506–521, 1995. 2. T. Anand, S. McCord, C. Sharma, and R. Balachander, “An integrated machine vision based system for solving the nonconvex cutting stock problem using genetic algorithms,” Journal of Manufacturing Systems, vol. 18, no. 6, pp. 396–415, 1999. 3. C. Bounsaythip, S. Maouche, and M. Neus, “Evolutionary search techniques: Application to automated lay-planning optimisation problem,” IEEE international conference on Systems, Manufacturing and Cybernetics (SMC), Vancouver, Canada, 1995, vol. 5, pp. 4497–4503. 4. A. Kahng, “Classical Floorplanning harmful?” in International Symposium on Physical Design, Proceeding ACM/IEEE, 2000, pp. 207–213. 5. P. Clay and A.J. Crispin, “Automated lay-planning in leather manufacturing,” in 17th National Conference on Manufacturing Research, Cardiff, 2001, pp. 257–262, ISBN 1-86058-325-3. 6. M. Adamowicz and A. Albano, “Nesting two-dimensional shapes in rectangular modules,” Computer-Aided Design, vol. 8, no. 1, pp. 27–33, 1976. 7. E. Burke and G. Kendall, “Evaluation of two-dimensional bin packing using the no fit polygon,” in Computers and Industrial Engineering Conference, Melbourne, Australia, 1999. 8. R. Cunninghame-Green, “Geometry, shoemaking and the milk tray problem,” New Scientist, vol. 12, 1677, pp. 50–53 1989. 9. P.K. Ghosh, “A unified computational framework for Minkowski operations,” Computer and Graphics, vol. 17, no. 4, pp. 119–144 1993. 10. J.A. Bennell, K.A. Dowsland, and W.B. Dowsland, “The irregular cutting-stock problem—a new procedure for deriving the no-fit polygon,” Computers and Operations Research, vol. 28, pp. 271–287, 2000. 11. A.J. Crispin, P. Clay, G.E. Taylor, R. Hackney, T. Bayes, and D. Reedman, “Genetic Algorithms applied to material utilisation in leather shoe manufacturing,” in 18th National Conference on Manufacturing Research, Leeds, 2002, pp. 267–272. ISBN 186058-378-4. 12. J. Heistermann and T. Lengauer, “The nesting problem in the leather manufacturing industry,” Annals of Operations Research, vol. 57, pp. 147–173, 1995. 13. D.E. Goldberg, Genetic Algorithms in Search, Optimisation and Machine Learning, Addison Wesley, Longman Inc. 1989, ISBN 0-020-15767-5. 14. M. Mitchell, An Introduction to Genetic Algorithms, Massachusetts Institute of Technology Press, 1996, ISBN 0-26213316-4. 15. P. Clay, A.J. Crispin, and S. Crossley, “Comparative analysis of search methods as applied to shearographic fringe modelling,” in Proceeding of 13th International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, New Orleans, June 2000, pp. 99–108. 16. B. Brassard and P. Bratley, Fundamentals of Algorithmics, Prentice-Hall: New Jersey, 1996, ISBN 0-13-335068-1. 17. J. Gross and J. Yellen, Graph Theory and Its Applications, CRC Press: Florida, 1998, ISBN 0-8493-3982-0. 18. H. M¨uhlenbein and D. Schlierkamp-Voosen, “Predictive Models
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for the Breeder Genetic Algorithm,” Evolutionary Computation, vol. 1, no. 1, pp. 25–49, 1993. 19. A.J. Crispin, P. Clay, G.E. Taylor, R. Hackney, T. Bayes, and D. Reedman, “Genetic algorithm optimisation of part placement using a connection based coding method,” in 15th International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, Cairns, Australia, 2002.
Alan Crispin is a Principal Lecturer in the School of Technology at Leeds Metropolitan University. His current research interests have focussed on component lay-out optimisation using genetic algorithms and other heuristics. He is a Chartered Engineer and obtained his Ph.D. from the University of East Anglia. He has authored one book and has published 40 papers.
Paul Clay is a doctoral student currently working as Product Development Manager at Pulsonic Technologies Ltd. His Ph.D. studies involve the application of heuristic optimisation to leather nesting. He received his B.Eng. (Hons) in Electrical and Electronic Engineering from Leeds Metropolitan University.
Gaynor Taylor has recently retired as Deputy Vice-Chancellor at Leeds Metropolitan University. She has been involved in robotics and automation since the early eighties when she was a member of the Robotics Research Group at Hull University. She received a B.Sc., M.Sc. and Ph.D. from UMIST.
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Tom Bayes is team leader of the Advanced Concepts section within the Shoe and Allied Trade Research Association (SATRA). He has been working on the problem of leather cutting for several years having developed one of the first computer-aided nesting systems to meet industry requirements.
David Reedman, recently deceased, was formally the research manager of British United Shoe Machinery with particular expertise in automatic control and system engineering. He had been working in the shoe industry for over 30 years.
