I Jornadas de Classificação e Análise de Dados Associação Portuguesa de Classificação e Análise de Dados Lisboa, 1 a 3 de Abril de 2004
Application of Clustering Methods for Optimizing the Location of Treated Wood Remediation Units Helena Gomes1*, Victor Lobo1,2, Alexandra B. Ribeiro3 1 Instituto Superior de Estatística e Gestão de Informação, Universidade Nova de Lisboa, * E-mail:
[email protected] 2 Escola Naval (Portuguese Naval Academy) 3 Departamento de Ciências e Engenharia do Ambiente, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa
Sumário: Foram utilizados dois métodos de análise de clusters (SOM e K-means) para optimizar a localização de unidades de remediação de resíduos de madeira preservada com CCA em Portugal. O software utilizado foi o SOM Toolbox para Matlab com dados de população dos Censos 2001 do Instituto Nacional de Estatística. Os dois métodos utilizados permitiram obter bons resultados, mas concluiu-se que, para este problema em particular, o K-means permitiu obter melhores soluções. Abstract: In this paper we used two clustering methods (SOM and K-means) to optimize the location of CCA-treated wood waste remediation units in Portugal. We used the SOM Toolbox for Matlab implementation with the population data from the last census made by the National Statistics Institute. The two methods yield good results, but we concluded that the kmeans algorithm provided better solutions than the SOM in this particular problem.
Key words: Self-Organizing Map (SOM), k-means, p-median problem, CCA-treated wood waste
1.
INTRODUCTION
In order to increase the service life of wood and to provide long-term resistance to attack by fungi, insects and marine organisms, wood is submitted to preservation techniques that generally apply wood preservatives using pressure. The wood preservative CCA (Chromated Copper Arsenate) is actually the most used wood preservative worldwide. Examples of treated wood products are telephone and utility poles, railway ties, fence posts and lumbers for outdoor construction (house fronts, roof tiles, window frames, playing tools, garden houses, fences, landings, wharves, bridges, bank revetment, acoustic barriers, etc). There is growing concern about the environmental impacts and increasing difficulty to dispose preservative treated wood products at the end of their service life. In the next decades a significant increase is expected in the amounts of treated wood removed from service. For example, in the USA the total volume of treated wood
removed from service was estimated to be near 8 million m3 in 1990, 9 million m3 in 2000, 15 million m3 in 2010 and about 18 million m3 in 2020 (McQueen et al., 1998). Consumer lumber makes up the majority of current and future expected production of CCA treated wood – about 56% of 1995 production compared to about 82% of expected waste wood generated in 2010 (Cooper, 2003). The future of the wood preservation industry depends on the solution(s) found to manage the treated wood waste problem. The recycling of these wastes, containing chromium, copper and arsenic (in the case of CCA-treated wood), can only be made after its remediation, so that planning and optimization of remediation units and its locations is of major importance. The objective of this study is the application of clustering methods to optimize the location of remediation plants for the treatment of preserved wood waste for further recycling, minimizing costs and respecting environmental criteria. Essentially, this is a p-median problem, a classic optimization problem that involves the location of facilities in such a manner that the total weighted distance of all users to their closest facility is minimized. The p-median problem has been proved to be NP-complete (Megiddo & Supowit, 1984; Plastia, 2002), and thus a globally optimal solution is very had to obtain. Different heuristics have been use to solve the p-median problem, such as k-means and k–medoids (Estivill-Castro & Houle, 2001); genetic algorithms (Correa et al., 2001); local search heuristics (Lorena & Sene, 2002); tabu search and variable neighbourhood search (Crainic et al., 2001; Mladenovic´ et al., 2003); simulated annealing (Church & Sorensen, 1994); Greedy Randomised Adaptive Search Procedure (GRASP) (Resende & Werneck, 2002); Branch-and-Cut algorithm (Avella et al., 2003) and Lagrangean Relaxation (Senne & Lorena, 2000). In this paper we will solve our particular instance of the p-median problem with a Self-Organizing Map (Kohonen, 2001), and k-means algorithm (MacQueen, 1967).
2. DATASET USED P-median problems are usually formalized by considering a number of demand nodes, each with a given location and weight (amount of resourced demanded), a number of facility locations that must be found, and a cost function that takes into account the relation between the demand nodes and the facility locations. In our case, the demand nodes will be the producers of waste wood, the facilities will be the remediation plants, and in a first approximation, the cost function will be the Euclidean distance between the demand nodes and the facilities, weighed by the amount of wood generated at each demand node. In this paper we assume that all wood that needs to be sent to the remediation plants is generated by municipal solid waste, which in turn is proportional to the number of inhabitants in each area. Thus we use population instead of amount of wood needing
remediation. A GIS (Geographic Information System) based location model was used to find the population of each “freguesia” of Portugal (NUTS 5). The data contained in that GIS comes from the latest census data (from the year 2001), provided by the Portuguese Statistics Institute (INE), and consists of the geographical location of the centroid of the “freguesia” and its population. The implementations of the clustering techniques that we used did not allow weighing of different data patterns. To achieve this weighing we generated one data pattern for each 1000 inhabitants of each “freguesia” (using excess rounding). We were then left with 99.146 data patterns. These data patterns were then put in random order so as not to introduce unnecessary bias. In all our tests we assumed that 6 remediation plants would be necessary.
3. PROPOSED SOLUTIONS Two kinds of clustering methods were used: the Self-Organizing Map (SOM) and kmeans algorithm. In our experiments we used the SOM Toolbox for Matlab implementation developed at Helsinki University of Technology (Vesanto et al., 2000). The SOM provides a visual representation of a vector quantification algorithm that places a number of vectors into a high-dimensional input data space in such a way that they approximate the original data patterns in an ordered fashion (Kohonen et al., 1995). It’s an unsupervised neural network that tries to preserve topological relations between points in the input and output spaces. Generally, SOMs have many neurons and these are spread in the input space proportionally to the density of data points. When we use very few neurons, as is the case in our experiments, the strong coupling between neighboring neurons will prevail over the minimization of the distances to the data patterns, thus introducing some distortions to our original problem. The k-means algorithm (MacQueen, 1967) is one of the classic statistical clustering methods. It generates a specific number of disjoint, flat (non-hierarchical) clusters. The k-means method is numerical, unsupervised, non-deterministic and iterative, and is basically a stochastic hill-climbing optimization technique. One of the main drawbacks in using k-means is that it is quite sensitive to local minima, and to a certain degree we verified that in our tests. There are several ways of dealing with this problem, the most common of which is to re-initialize the kmeans algorithm several times with different seeds, and then choose the best solution.
4. EXPERIMENTAL RESULTS When using SOM, we have to provide an output space grid upon which the neurons are set. In our experiments we used two different grids, a 2x3 and a 1x6 grid. For the 2x3 grid we used both a hexagonal and a rectangular topology. In all three experiments, we used random initialization, and two learning phases. In the first phase we used an initial radius of 3 (linearly decreasing to 0), a learning rate of 0.3 (also decreasing linearly to 0), a sequential training algorithm, 3 epochs of training and bubble neighborhood. The second learning phase differed from the first in the initial radius (we used an initial radius of 2), in the learning rate (0.05) and in the number of training epochs (6). For the k-means algorithm, we defined six centroids and 2400 training epochs, using a batch algorithm. We did 4 different experiments with this method, using the same parameters. The results are presented in Figure 1. The average distance to a remediation plant in each of the solutions is given in table 1. Table 1- Average distance (in km) to a remediation plant K-means (1st run) 32.779 SOM (2x3 hex.) 50.724 K-means (2nd run) 32.778 SOM (2x3 rect.) 51.576 K-means (3rd run) 41.510 SOM (6x1.) 44.730 K-means (4th run) 32.526 From these results we can see that the SOM introduces a tight coupling amongst the locations of the plants, forcing them to be close to each other, and thus increasing the total cost of the solution. It is also clear that the k-means algorithm is reasonably robust to initial conditions (only one solution is considerably different to the other three), and yields good solutions to the problem.
5. CONCLUSIONS In this paper we show that the choice of the best locations for wood remediation plants can be found using the k-means algorithm, as well as the SOM. The k-means algorithm provided better solutions than the SOM in this particular problem. This can be explained by the fact that SOMs are better suited to problems where the number of units to be placed is large, or when there is a need to model a forced proximity between these units. As this is not the case, we suggest that k-means be used to solve the wood remediation plant location problem. The solutions obtained with our data make sense and could be used to decide on the location of these plants.
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c) Final map with SOM results (x) and k-means results (o).
Figure 1 – Results of the application of clustering algorithms.
The approach used has great potential because it can easily be adapted to a more complex (and realistic) formulation of our problem. On one hand, additional sources of preserved wood waste can be added. Each source of preserved wood (population, industrial facilities, agriculture, etc) can be described as pattern composed of geographical coordinates and a quantity of treated wood that acts as weight. On the other hand, both algorithms used can use distances derived from a distance matrix with actual road distances between possible locations, thus turning the problem more realistic. Further developments of this work include: i) the identification of the key environmental and economic factors to be considered in the optimization model, ii) the process of formulation and combination of these criteria; iii) analysis and comparison of heuristics in the p-median problem resolution; iv) production of a final map with the visualization of areas suitable for the location of remediation infrastructures.
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Mladenovic´, N. et al. (2003). Solving the p-Center Problem with Tabu Search and Variable Neighborhood Search. NETWORKS, 42,(1): 48–64. Plastia, F. (2002). Continuous Covering Location Problems. Facility Location Applications and Theory. Z. Drezner & H. W. Hamacher, Springer-Verlag, Heidelberg. Resende, M. G. C. & R. F. Werneck (2002). A GRASP with Path-relinking for the PMedian Problem. Technical Report TD5E4QKA, AT&T Labs Research.22. Senne, E. L. F. & L. A. N. Lorena (2000). Lagrangean/surrogate heuristics for pmedian problems. Computing Tools for Modeling, Optimization and Simulation: Interfaces in Computer Science and Operations Research. M. L. a. J. L. GonzalezVelarde, Kluwer Academic Publishers: pp 115-130. Vesanto, J. et al. (2000). SOM Toolbox for Matlab 5. Helsinki University of Technology.
