Fuzzy Stopping Rules for the Strip Packing Problem

Transcription

1 Fuzzy Stopping Rules for the Strip Packing Problem Jesús David Beltrán, Jose Eduardo Calderón, Rayco Jorge Cabrera, José A. Moreno Pérez and J. Marcos Moreno-Vega Departamento de E.I.O. y Computación Escuela Técnica Superior de Ingeniería Informática, Universidad de La Laguna La Laguna SPAIN. jamoreno@ull.es, jmmoreno@ull.es Abstract The stopping rule is one of the elements of the metaheuristics least studied. One of the motives is, surely, the imprecision in the terminology used by experts to formulate the stopping rules. In this work we consider the formulation of stopping rules in fuzzy terms and analyze their application to the Strip Packing Problem. We experimentally compare two kind of stopping rules: dependent and independent of the problem, and compare them. Keywords: Metaheuristics, Fuzzy Stopping Rules, Strip Packing Problem 1 Introduction Metaheuristic searches are general solution procedures for solving problems. They are characterized by their wide applicability and by their good performance in many cases. Thus, ideally, they provide high quality solutions consuming moderate resource; time. In general, the concepts of high quality solutions and moderate resource consuming are subjective and, therefore, appropriately modelled by fuzzy sets. The Fuzzy sets-based heuristic FANS [3], [4] is the first metaheuristic that incorporate fuzzy sets and it includes the use of a fuzzy or subjective valuation of the solutions, that is one of its most relevant elements. The Neighborhood Metaheuristic search procedures constitute a important kind of metaheuristics. These metaheuristics use neighborhood structures in the solution space. A neighborhood structure is a mapping N which defines for each solution x a set of solutions that are close to x in some sense. These solutions are the neighbor solutions of x and constitute its neighborhood N(x). In the search, some moves from a current solution to a neighbor solution are executed, until a stopping condition is met. The mechanism applied to generate and accept the neighbors of the current solution determines the metaheuristic. This mechanism can be as simple as generating a random neighbor and always accept it, or a sophisticated improving procedure. In any case, the search must stop when the solution can not be improved with reasonable resource consuming. This must be achieved by the stopping criterion used in the search. The characteristics of the solutions examined can be analyzed to stop the procedure when they indicate that it is difficult to improve the best solution found. In this work we describe the use of fuzzy sets to design stopping criteria for the metaheuristic searches. We formulate fuzzy stopping criteria that are general and specific for the Strip Packing Problem (SPP) and compare them. Consider a strip of fixed width w and infinite height, and a finite set of rectangles, with at least one of their sides less than w. The SPP consists of packing the rectangles in the strip minimizing the height of the packing; see figure 1. In this problem the rectangles can be

2 neighborhood of the current solution. If this neighbor is better than the current solution, the process continues with the neighbor as the current solution. Otherwise, the process stops with the current solution, that is a local optimum, as proposed solution for the problem. In the Multi-Start Search (MSS) [10], local searches from several initial solutions are executed until the stopping condition is met. The best local optimum found by the local searches is the proposed solution for the problem. 2.2 Variable Neighborhood Search Figure 1: Solution of a Strip Packing Problem, SPP). rotate at 90 degrees and the cuttings can be non guillotine. A cutting is guillotine if it goes from a side of the object to the front side. In a non-guillotine cut this can not be true. The rest of the paper is structured as follows. In the next section we describe the Metaheuristic searches that will be used in the computational experience. In the section 3 we formulate the Strip Packing Problem and fuzzy stopping rules for this problem are described in section 4. In section 5 we do an experimental comparative study of the fuzzy stopping rules for the metaheuristic searches to solve the problem. Finally, in the section 6 we enumerate the conclusions of our cumputational experience. 2 Metaheuristic Searches The metaheuristc searches generate a series of solutions of the problem to provide the best solution found. The stopping rules determine when to stop the search. We now describe the metaheuristics that will be used to compare the stopping rule defined in Multi-Start Search A local search starts with a random initial solution as current solution. At each iteration, it determines the best solution of the The Variable Neighbourhood Search (VNS) [6], [7] uses a series of neighborhood structures N k, k = 1,..., k max defined for each solution. Initially it takes k = 1 and a random initial solution as current solution. A local search from the current solution with the neighborhood N 1 is executed to return a local optimum x. The following steps are repeated until the stopping condition is met. Take k = k + 1 and a random neighbor y of the local optimum x using in the neighborhood structure N k ; i.e., y N k (x). Apply a local search from y using the neighborhood structure N 1. If the local minimum found is better than x then the process is iterated with this new local minimum as current solution x and k = 1. Otherwise, do k = k + 1 and iterate the process with the same current local optimum x. When k = k max, generate a new initial solution x and iterate the process with k = GRASP The Greedy Randomized Adaptive Search Procedure (GRASP) [12] has two phases: a constructive phase and an improving phase that is applied to the result of the constructive phase. The improving phase can be any local search and the constructive phase consists in the following. Consider a heuristic function h that measures how convenient is the inclusion of the element as part of the solution. Starting with an initial empty solution, at each iteration, the procedure the procedure adds a new element to the partial solution by

3 the following steps. Determine the value of function h for all the possible new elements. Build a restricted candidate list (RCL) with the best elements according to h. In general, in the RCL are included the m (a parameter fixed by the user) best elements. Select a random element of RCL and include it in the partial solution. Repeat this process until a complete solution of the problem is built. The constructive and improvement phases are repeated until the stopping criterion is met. 3 The Strip Packing Problem The Strip Packing Problem (SPP) is formulated as follows. Let w be the width of the strip with infinite height and let R = {R(w i, h i ) : i = 1,..., n} be a set of n rectangles. Each rectangle R(w i, h i ) has width w i and height h i verifying min{w i, h i } w, for each i = 1,..., n. The problem is to establish the optimal placing (r i, a i, b i ) of the rectangles R(w i, h i ), i = 1,..., n. Here r i is a binary variable that represents if the i-th rectangle is rotated or not, and (a i, b i ) are the coordinates of the position of the bottom-left corner of the rectangle R(w i, h i ) with respect to the origin of coordinates (this origin is fixed at the bottom-left corner of the strip). If r i = 1 then the i-th rectangle is rotated at 90 degrees; otherwise r i = 0 and it is not rotated. The placing of each rectangle is feasible if the following two conditions are verified: the rectangle is completely included in the strip and it does not overlap with any other rectangle already placed. Therefore, If r i = 0 then 0 a i w w i, 0 b i and the position of no other rectangle R(w j, h j ) verifies a i < a j < a i + w i and b i < b j < b i + h i. If r i = 1 then 0 a i w h i, 0 b i and the position of no other rectangle R(w j, h j ) verifies a i < a j < a i + h i and b i < b j < b i + w i. The objective function to be minimized is the maximum height h reached by the rectangles that is computed by: h = max{max r i =0 (b i + h i ), max r i =1 (b i + w i )} Some infinite set of equivalent feasible solutions could be obtained by shifting horizontally or vertically some rectangles when it is possible. To avoid it, only solutions obtained by introducing successively all the rectangles following a given placement strategy are considered. We use the following usual bottomleft strategy: each rectangle is placed at the deepest location and, within it, in the most left possible location. So, each feasible solution of the problem is determined by the order in which the rectangles are introduced in the strip. Therefore, the space of solutions consists of the permutations of the numbers [1,..., n] with a binary vector [r 1,..., r n ] that represents if each rectangle is rotated or not. The solution represented in figure 2 is obtained by introducing the five rectangles labelled with numbers 1 to 5 in this order, following the above strategy. In this solution, it is possible that some rectangles have been rotated at 90 degrees. The figure shown may be the configuration corresponding to a partial solution where some rectangles are not placed yet. There we can see two important elements: the space closed by the rectangles 2, 3 and 4, that is an area wasted, and the shape of the upper contour of the set of rectangles, that constraints the possible location of the next rectangles. Observe that, if it is a complete solution, the area between the upper contour and the maximum height reached by a rectangle is also wasted see figure 3. 4 Fuzzy Stopping Rules One of the elements that most influence has in the success of the search procedures to solve a problem is the stopping criterion applied. The stopping rules must provide an appropriated trade-off between efficiency and efficacy of the solution procedure by detecting when a hight quality solution is reached. In [13] and [14] we found the first fuzzy stopping rules appeared in the literature that

4 are used for exact algorithms to solve the Knapsack and Travelling Salesman Problems. Those rules are based on the fact that most of the decision makers would accept solutions that are not optimal but good enough. They use membership functions depending on the objective to establish the set of good solutions. The corresponding α-cuts determine general and robust stopping criteria. The stopping rules, like the solution procedures, can be general or specific. From the point of view of the field of Metaheuristics, where the generality of the procedures is one of the most relevant desirable characteristics, the stopping rules must also be general. However, as we show in this work, the stopping rules depending of the problem are more efficient and effective. The general or independent of the problem stopping rules are those which can be applied to any optimization problem. Some general stopping rules are those included in the cooling schedule of the Simulated Annealing [1] and those found in [5] and [9]. Basic general stopping rules are those based on the computation time, number of iterations or number of times an basic procedure is applied (for instance, number of solution evaluations). These quantities are alternative ways to measure computational resource. The stopping criterion is to stop when this number is large enough. We propose the fuzzy stopping rule obtained by considering the set of large enough numbers defined by the membership function: µ(x) = x k, where k is a parameter provided by the decision maker. The main drawback of the general rules is that they are not efficient, since they do not use additional information. Better stopping rules are obtained by taking into account characteristics of the procedures applied, the evolution of the search process, evaluations of the objective and other characteristics of the solutions found like it is done in the FANS metaheuristcis [3], [4]. The problem dependent stopping rules use (y 1, x 1 y 1 1, x1 2 ) y 3 y 2 5 (y 2, x 2 1, x2 2 ) (y 3, x 3 1, x3 2 ) x 1 1 x 1 2 = x2 1 x 2 2 = x3 1 x 3 2 Figure 2: Upper contour of a partial solution with 5 packed rectangles properties or characteristics of the problem or its solutions. We propose stopping rules for the Strip Packing Problem by employing the property of solutions difficult to improve. This property is established as a fuzzy set with the corresponding membership function. In general, the quality of a solution for a problem is evaluated taking into account only the objective value. However, there are other aspects of the solution that indicate the possibilities of improving it. In the SPP, the set of wastes and the shape of the upper contour are two of these characteristics. The properties that indicate how must be these characteristics for high quality solutions are naturally fuzzy. Solutions for the SPP with a small waste and a smooth contour are high quality solutions. To evaluate the waste we can use the number of areas wasted or their total area. The smoothness of the contour can be evaluate by several measures of the variability among the heights of the segments that constitutes the upper contour or by the area between the upper contour and the maximum height. Note that this area is proportional to the weighted average of the difference between the maximum height and the height of the segments of the upper contour, where the weights are the width of these segments. In [11] we use one of these characteristics to define a fuzzy stopping rule. In the solution represented in figure 2 the upper contour are identified for the segments that constitute it from left to right. There-

5 fore it can be represented by the following sequence of tuples C = [(y i, x i 1, xi 2 ) : i = 1,..., c], where y i is the height of the i-th segment and x i 1 and xi 2 its initial and finals. This sequence, even if it represents the upper contour a partial solution, verifies x 1 1 = 0, xc 2 = w and x i+1 1 = x i 2, for each i = 1,..., c 1. Let X be a solution with upper contour C. The value of the objective function is the maximum height: h 5 W w W h = f(x) = max i=1,...,c {yi }. The smoothness of the contour C is measured by the difference between the maximum height h and the weighted average of the heights y i of the segments that compound it, where the weight of the i-th segments is its length; i.e. x 2 i x1 i. This value is computed by: S(X) = f(x) 1 c y i (x 2 i x 1 i ) w i=1 = 1 c (f(x) y i )(x 2 i x 1 i ). w i=1 Note that S(X) is w times the sum of the rectangles with width equal to the length of each segment and height equal to the corresponding distance to the maximum height h. The waste W (X) of a solution X is the total area of the wasted zones. This includes the areas wasted among packed rectangles and also the area wasted between the upper contour and the maximum height. If A denotes the total area of the rectangles to be packed, n A = w i h i, i=1 then W (X) = w h A (see figure 3). We use the fuzzy concepts of small waste and smooth contour to define fuzzy problemdependent stopping rules. Let λ 1 [0, 1] be a value fixed by the decision maker. The set of solutions with smooth contour is given by the membership function µ 1 given by: µ 1 (X, λ 1 ) = 1 λ 1 S(X) A Figure 3: Total Waste of a solution with 5 packed rectangles Thus, the membership function of a solution is 1 if S(X) is 0, and decreases linearly when S(X) increases. A first fuzzy stopping rule stops the search when the process finds a solution with smooth contour, for the decision maker. Similarly, let λ 2 [0, 1] be another value fixed by the decision maker. The set of solutions with small waste is given by the membership function µ 2 given by: µ 2 (X, λ 2 ) = 1 λ 2 W (X) A A second fuzzy stopping rule stops the search when the process finds a solution with small waste. A third fuzzy stopping rule combines both sets of solutions by stopping the search process when it finds a solution with few waste and smooth contour, for the decision maker. In the practical implementations, the decision maker fixes the level α for the α-cut. Thus the search stops when W (X) 1 α A λ 2 The concrete values for the parameters that appear in the stopping rules can be fixed together. For instance, to say that the percentage of waste must be less 5% of the total area A of the rectangles we set: (1 α)/λ 2 = Computational Experience To analyze the performance of the fuzzy stopping rules we solve instances of the

6 Table 1: Pure Random Search (n iter 1000) 0.20% 0.15% 0.10% 0.05% Indep. Cat. n iter h n iter h n iter h n iter h n iter = 3000 C (16) C (26) C (26) C (26) C (10) C (5) Table 2: Multi Start (n ls 1000) 0.20% 0.15% 0.10% 0.05% Indep. Cat. n ls h n ls h n ls h n ls h n ls = 10 C (19) C C (17) C C (21) C (19) Strip Packing Problem with the metaheuristic searches enumerated in section 2. As usual in the searches in solution spaces consisting of permutations, the random generation of solutions is performed by successively selecting one of the remainder elements with the same probability. The usual move in neighbor-based metaheuristics is the interchange of the position of two elements of the permutation. The neighborhood of a solution consists of the solutions obtained by a move. This neighborhood structure is used in the local searches. The multiple neighborhoods for the VNS are obtained by nesting them by the iteration of the above move. The details of the GRASP applied in our computational experience are described in [2]. The problem-independent stopping rules can be based in several counters or quantities; number of iterations, number of solution evaluations, CPU time, etc. We use different numbers as counter for the four metaheuristics considered. The counter is the number of iterations for the PRS denoted by n iter and the number of local searches denoted by n ls for the MSS. For the VNS we use as counter the number n imp improvements; i.e., the number of times that k reaches the k max value that is fixed to 3. Finally for the GRASP the counter is the number of of constructive phases n con. The instances used are those corresponding to the categories C 1, C 2,..., C 6 of Hopper and Turton [8] (available at: mscmga.ms.ic.ac.uk/jeb/orlib/stripinfo.html). Each category consists of 3 instances with the following characteristics. The instances in C 1 have n = 16 or 17 rectangles, the strip has width w = 20 and the optimal height is h opt = 20. The category C 2 consists of 3 instances with n = 25 rectangles, w = 40 and h opt = 20. The instances of C 3 have 28 or 29 rectangles, width w = 20 and optimal height h opt = 20. The three instances of C 4 have 49 rectangles and width and the optimal height are equal to 60. Category C 5 consists of 3 instances with n = 72 and 73, w = 60 and h opt = 90. Finally, the last category C 6 consists of 3 instances with n = 97 rectangles, the strip has width w = 40 and the height of the optimal solution is h opt = 20. Tables 1, 2, 3 and 4 show, respectively, the results obtained with PRS, MSS, VNS and GRASP metaheuristics. In the first column

7 Table 3: VNS(n imp 1000) 0.20% 0.15% 0.10% 0.05% Indep. Cat. n imp h n imp h n imp h n imp h n imp = 5 C C C C C C Table 4: GRASP(n con 1000) 0.20% 0.15% 0.10% 0.05% Indep. Cat. n con h n con h n con h n con h n con = 10 C (28) (12) C (10) C (29) (19) C (24) C C is the category of instance in the Hopper and Turton repository. We consider the fuzzy stopping rule by fixing four values for the percentage of total waste W (X): 0.20%, 0.15%, 0.10% and 0.05%. The next four pairs of columns of the table show the corresponding counter and the objective value when the solution obtained reach each one of these percentages. They are average values on the 3 instances in each category that have been solved 10 times with each metaheuritic. However, we stop the search when the counter is 1000 even if it has not reach the smallest percentage. Therefore, in several cases, the average values are obtained for less than 30 executions (this number is indicated between round brackets). The last column indicates the average objective function obtained with the independent of the problem stopping rule that uses the corresponding counter. 6 Conclusiones We obtain the following conclusions from the analysis of the computational result: 1. The problem dependent and problem independent stopping rules are adequately formalized using fuzzy sets. 2. The problem dependent fuzzy stopping rule was more effective than the problem independent stopping rule for the same problem. With the problem independent stopping rules the executions of the same metaheuristics for the problems in the same category obtain very different objective values, even for the same problem, but with similar computational time. However, with the problem dependent stopping rules, the executions of each metaheuristics for the problems in each category obtain similar objective values but with different computational time. 3. As better is the metaheuristic more recommendable is the use of the problem dependent fuzzy stopping rule. In heuristics that provide low quality solutions, like the Pure Random Search, is unlike to find a solution that fulfill the stopping condition of the problem dependent fuzzy stopping rule. The opposite occurs for the GRASP, that can find solutions that fulfills the stopping condition of the

8 problem dependent fuzzy stopping rule in very few iterations. From the trade-off between the wasted areas among the rectangles and the wasted area between the upper contour and the maximum height it seems that for the high quality solutions, the interior waste is more important than the superior waste. This would be analyzed in a forthcoming research. Acknowledgements This work has been partially supported by the Spanish Ministry of Science and Technology through the project TIC C03-01; 70% of which are FEDER founds. References [1] Aarts, E, Korst, J. Simulated Annealing and Boltzmann Machines. John Wiley and Sons (1989) [2] Beltrán, Jesús David; Eduardo Calderón, Jose; Jorge Cabrera, Rayco; Moreno Vega, J. Marcos Procedimientos constructivos adaptativos (GRASP) para el problema del empaquetado de rectángulos Inteligencia Artificial. Revista Iberoamericana de Inteligencia Artificial (2002) 15, [3] Blanco, A., Pelta, D., Verdegay, J.L. A Fuzzy Valuation-based Local Search Framework for Combinatorial Problems. Fuzzy Optimization and Decision Making. (2002) 1, [4] Blanco, A., Pelta, D., Verdegay, J.L. Applying a Fuzzy Sets-based Heuristic to the Protein Structure Prediction Problem. International Journal of Intelligent Systems. (2002) 17, [5] Boender, C.G.E., Rinnooy Kan, A.H.G. Bayesian Stopping Rules for Multistart Global Optimization Methods. Mathematical Programming (1987) 37, [7] Hansen, P., Mladenovic, N.: Variable neighborhood search: Principles and applications. European Journal of Operational Research. (2001) 130, [8] Hopper, E., Turton, B.C.H.: A Review of the Application of Meta-Heuristics Algorithms to 2D Strip Packing Problems. Artificial Intelligence Review (2001) 16, [9] Los, M., Lardinois, C.: Combinatorial Programming, Statistical Optimization and the Optimal Transportation Network Problem. Transportation Research (1982) 2, [10] Martí, R.: Multi-Start Methods Handbook of MetaHeuristics, Glover and Kochenberger (Eds.), Kluwer (2003), [11] Moreno Pérez, José A., Moreno-Vega, J. Marcos: Fuzzy Constructive Heuristics. En Fuzzy Sets Based Heuristics for Optimization, José L. Verdegay (Ed.). Springer Verlag. Series: Studies in Fuzziness and Soft Computing, Volumen. 126 (2003) [12] Resende, M.G.C., González Velarde, J.L.: GRASP: Procedimientos de búsquedas miopes aleatorizados y adaptativos. Inteligencia Artificial. Revista Iberoamericana de Inteligencia Artificial. (2003) 19, [13] Sancho-Royo, A., Verdegay, J.L., Vergara-Moreno, E.: Some Practical Problems in Fuzzy Sets-based Decision Support Systems. MathWare and Soft Computing (1999) Vol. VI, N. 2-3, [14] Verdegay, J.L., Vergara-Moreno, E.: Fuzzy Termination Criteria in Knapsack Problem Algorithms. Mathware and Soft Computing. (2000) Vol. VII, N. 2-3, [6] Hansen, P., Mladenovic, N.: Variable neighborhood search. Computers & Operations Reserach. (1997) 24,