A GRASP Approach to the Nesting Problem

Transcription

1 MIC th Metaheuristics International Conference 47 A GRASP Approach to the Nesting Problem António Miguel Gomes José Fernando Oliveira Faculdade de Engenharia da Universidade do Porto Rua Roberto Frias, Porto - Portugal {agomes, jfo}@fe.up.pt INESC Porto - Instituto de Engenharia de Sistemas e Computadores do Porto Rua José Falcão 110, Porto - Portugal 1 Introduction The nesting problem is a two-dimensional cutting and packing problem where the small pieces to cut have irregular shapes. The variant of the problem that we are going to deal with, which naturally arises in several industrial production processes (textile, garment, metalware, etc), considers only one big rectangular piece, the plate, with fixed width and infinite length. The objective will be the minimisation of the length of the plate that is used to produce a given set of small pieces. The nesting problem is also characterised by the intrinsic difficulty of dealing with geometry, once pieces are usually represented by non-convex polygons. The satisfaction of the no overlapping and containment constraints involve complex computations. Several approaches have been proposed for the resolution of nesting problems. Solution techniques rangefromsimpleheuristicstolocaloptimisationtechniques[2,4,5,6,9,10,11,13,14,15].evenfor small instances the mathematical programming model of the problem is not solvable in a reasonable amount of time. In this paper, we propose a GRASP[7, 8] (Greedy Randomised Adaptive Search Procedure) approach to tackle the nesting problem. The idea behind this approach is to represent a solution by a sequence of pieces and perform the search over this representation, exchanging pairs of pieces in the sequence. A low level placement heuristic is needed for generating the cutting patterns (layouts), one for each particular sequence of pieces. 2 Dealing with Geometry In any cutting or packing problem the layout geometric feasibility has to be tackled, i.e. pieces must not overlap each other and have to be completely inside the material to cut or the space to divide. In nesting problems, this geometric problem is particularly difficult due to the natural irregularity and non-convexity of the pieces to be placed. Since in this problem each solution evaluation implies building a layout, it is very important for the global nesting algorithm s efficiency to have an adequate geometry representation and manipulation. Pieces will then be represented by polygons. Arcs are approximated by a set of exterior tangents to the curve. Holes inside the interior of the pieces are not allowed. The concept of the no-fit-polygon is used to deal with the pieces no overlap restrictions. The problem of placing pieces inside the material to cut (the plate) is, in this particular case, easy to tackle once the plate is always a rectangle.

2 48 MIC th Metaheuristics International Conference NFP A,B IFR A,B A A R B B R B B (a) (b) Figure 1: The no-fit-polygon NFP A,B and the inner-fit-rectangle IFR A,B 2.1 The No-Fit-Polygon The concept of the no-fit-polygon was first introduced by Art [3], with this idea being used again by Adamowicz and Albano [1]. Later on Mahadevan [12] presented a comprehensive description of a no-fit-polygon algorithm implementation, which has been followed in the present work. The no-fit-polygon of piece B relative to piece A (NFP A,B ) is the locus of points traced by the reference point associated with B, when this piece slides along the external contour of A. Therelative orientations of A and B are kept fixed during this orbital movement. Piece B (the orbital piece) must never intersect A (the stationary piece) and they must always be in contact (figure 1(a)). From this definition it immediately follows that: If the reference point (R B ) of piece B is placed in the interior of NFP A,B then B intersects A. If the reference point (R B ) of piece B is placed on the boundary of NFP A,B then B touches A. If the reference point (R B ) of piece B is placed in the exterior of NFP A,B intersect or touch A. then B does not To achieve a feasible (without overlap) and tight layout, each piece should have its reference point on the boundary of at least one no-fit-polygon (relative to another piece) and in the exterior of all the other no-fit-polygons (relative to the remaining pieces). Directly from the concept of the no-fit-polygon, we can derive the concept of the inner-fit-rectangle (IFR A,B ), where the piece B slides along the internal contour of the rectangle B. Whenever exists, the IFR A,B is always a rectangle (figure 1(b)). 3 The GRASP Approach The GRASP algorithm is an iterative process, where each iteration consists of a constructive phase and a local search procedure. The constructive phase is based on a greedy heuristic that builds a feasible solution, whose neighbourhood is then explored by the local search procedure. At the end, the result is the best solution found over all of the iterations. The proposed GRASP approach deals with piece sequences and supposes a low level placement heuristic to actually place the pieces in the plate. This means that the search is conducted through a solution space of different sequences of pieces. The constructive procedure builds a sequence of pieces, adding one piece at a time, accordingly to a greedy selection criterion. This sequence is then converted

3 MIC th Metaheuristics International Conference 49 Pieces sequence Placement Heuristic Cutting pattern Figure 2: Placement heuristic. in a layout by a greedy bottom-left placement heuristic. The local search procedure is implemented by a 2-exchange mechanism between pairs of pieces in the current sequence. 3.1 Placement Heuristic The placement heuristic has to convert a sequence of pieces in a feasible layout (figure 2). It is a greedy bottom-left heuristic and the feasible placement points are the vertices of the no-fit-polygon NFP Pi,P k and the vertices of the inner-fit-rectangle IFR P,Pk and the intersection of two edges of two no-fit-polygons NFP Pi,P k and NFP Pj,P k and the intersection of an edge of the inner-fit-rectangle IFR P,Pk and an edge of a no-fit-polygon NFP Pi,P k. This means that piece P k will be in contact with the plate, with the plate and one or more of the pieces already placed, or with two or more of the pieces already placed. It should also be mentioned that the no-fit-polygon and the inner-fit-rectangle can both be computed off-line, since they only depend on the shape of the pieces. When different orientations are allowed, then for each pair of pieces it is necessary to calculate a no-fit-polygon for each pair of different orientations, and for each piece it is necessary to calculate a inner-fit-rectangle for each orientation. Fortunately, in real cases the admissible orientations are limited to a few possibilities due to technological constraints (drawing patterns, material resistance, etc). Another important characteristic of this heuristic is that holes, in the middle of the layout, can be filled at any stage of the placement, reducing in this way the dependence from the position of the pieces in the sequence. 3.2 Constructive Procedure The constructivephaseincorporatesthe greediness, the randomnessand the adaptiveness characteristics of the GRASP algorithm. The greediness is achieved by, at each iteration of the constructive procedure, ranking all the elements in one candidate list, according to a greedy function. This list is then restricted only to the elements which lead to good solutions, the restricted candidate list (RCL). Selecting one element randomly from the restricted candidate list, gives to GRASP the randomness characteristic: different solutions can be obtained. Finally, the adaptiveness is reached by updating, at each iteration of the constructive procedure, the benefits for selecting one element taking into account the previous elements. A parameter, α, controls how the candidate list is restricted, by imposing a threshold value on the distance to the best solution found (actually α is a percentage between the worst and the best solution found). This way, we can vary the GRASP algorithm from being completely greedy, α = 0, to being completely random, α = 1. In our approach, we have tried two different greedy functions to rank the candidate list. The first one was the layout length, measured as the maximum X coordinate in the current layout: in figure 3(a), placing piece A leads to a layout length l A, while placing piece B leads to a layout length l B. The second one was the added internal waste, measured as the potential area lost when placing one piece: in figure 3(b), placing piece C increases the internal waste by the area c 1 plus half of the area

4 50 MIC th Metaheuristics International Conference A B l a l b c 1 c 2 C c2 d 1 d 2 d 3 d 2 D (a) Layout length. (b) Internal waste added. Figure 3: Criteria to select the next piece in the sequence. =1 =2 =3 Figure 4: The neighbourhood size parameter. c 2 (because these areas can easily be used for a future placement), while placing piece D increases the internal waste by the area d 1 plus half of the area d 2 and minus the area d 3 (because this area was considered waste in a previous stage). Finally, different values for parameter α were also tried, from 0 to 1 (step 0.1). 3.3 The Local Search Procedure Applying a local search procedure, starting from a initial solution generated by the constructive phase should always be useful, since these solutions are not guaranteed to be locally optimal. The local search procedure used in this work is based on a 2-exchange mechanism procedure, which is responsible for guiding the search trough the solutions space. The algorithm moves to a neighbour solution by exchanging a pair of pieces in the current sequence. This is an improvement algorithm that ends when it is impossible to move to a better solution in the current neighbourhood. Neighbourhoods are build by exchanging pairs of pieces in a sequence. Parameter (figure 4) controls the size of the neighbourhood, allowing only exchanges between pieces within a distance of. For each piece, all the exchanges with the following pieces in the sequence area evaluated. Three different values for parameter delta were tried: = 1, 2, 3. Two different strategies were used to select the neighbour that will be the center of the neighbourhood in the next iteration, i.e. which pair of pieces will be exchanged: the first better solution found (first better); the best solution found (best ). The idea behind the first better strategy, when selecting the first better solution found in the neighbourhood, is to rapidly change from one solution to another. On the other hand, the best neighbour exhaustively searches the neighbourhood looking for the best solution.

5 MIC th Metaheuristics International Conference 51 (a) SHAPES0. (b) SHAPES1. (c) SHIRTS. (d) SHAPES2. (e) TROUSERS. Figure 5: Best layouts obtained with the GRASP approach. 4 Preliminary Results The GRASP approach was tested with 5 data sets available in the literature. Three of these data sets (SHAPES0, SHAPES1 and SHAPES2 ) are artificial, while the other two, SHIRTS and TROUSERS, are real instances taken from the garment industry. The actual data can be found in [14]. The proposed GRASP approach supposes a few tactical decisions, that must be taken to make the GRASP approach fully operational. These issues are: the size of the neighbourhood ( = 1, 2, 3), the strategy for searching the neighbourhood ( first better solution, best solution), the greedy function to rank the candidate list (layout length, added inner waste) and how much restricted the candidate list is (α [0, 1]). A set of preliminary experiments allowed us to decide on some of those issues: = 3 for the size of neighbourhood and the best solution found, as the strategy to search the neighbourhood, since in both cases the preliminary experiments clearly show that they outperform the others; Added inner waste as the greedy function to rank the candidate list, since the layout length function proved to be very insensible when placing one piece which do not increase the layout length (figure 3(a), piece A); no conclusions could be drawn on how the candidate list should be restricted further experiments must be performed. In figure 5 the best layouts obtained with the GRASP approach for each instance are represented. These results, despite being preliminary, shown the effectiveness of the GRASP approach when applied to nesting problems. The results presented here were obtained with only 10 GRASP iterations for each instance and nevertheless are quite good: they range 0.7% worse than the best known solution, for instance SHAPES2, to 10.0 % worse for instance TROUSERS.

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