Multidimensional Dynamic Programming Algorithm for -Level Batching with Hierarchical Clustering Structure

Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 6021708, 12 pages https://doi.org/10.1155/2017/6021708 Research Article Multidimensional Dynamic Programming Algorithm for 𝑁-Level Batching with Hierarchical Clustering Structure Seung-Kil Lim,1 June-Young Bang,1 and Jae-Gon Kim2 1 Department of Industrial and Management Engineering, Sungkyul University, 400-10 Anyang-dong, Manan-gu, Anyang-si, Gyeonggi-do 14097, Republic of Korea 2 Department of Industrial and Management Engineering, Incheon National University, 12-1 Songdo-dong, Yeonsu-gu, Incheon 22012, Republic of Korea Correspondence should be addressed to Jae-Gon Kim; Received 23 December 2016; Revised 3 March 2017; Accepted 6 March 2017; Published 11 April 2017 Academic Editor: Sergii V. Kavun Copyright © 2017 Seung-Kil Lim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This study focuses on the 𝑁-level batching problem with a hierarchical clustering structure. Clustering is the task of grouping a set of item types in such a way that item types in the same cluster are more similar (in some sense or another) to each other than to those in other clusters. In hierarchical clustering structure, more and more different item types are clustered together as the level of the hierarchy increases. 𝑁-level batching is the process by which items with different types are grouped into several batches passed from level 1 to level 𝑁 sequentially for given hierarchical clustering structure such that batches in each level should satisfy the maximum and minimum batch size requirements of the level. We consider two types of processing costs of the batches: unit processing cost and batch processing cost. We formulate the 𝑁-level batching problem with a hierarchical clustering structure as a nonlinear integer programming model with the objective of minimizing the total processing cost. To solve the problem optimally, we propose a multidimensional dynamic programming algorithm with an example. 1. Introduction According to Wikipedia, clustering problem is the task of grouping a set of item types in such a way that item types in the same cluster are more similar (in some sense or another) to each other than to those in other clusters. It is the main task of exploratory data mining and a common technique for statistical data analysis, used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, bioinformatics, data compression, and computer graphics. Hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis which seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two types: agglomerative clustering and divisive one. Agglomerative clustering is a bottom-up approach: that is, each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy. To the contrary, divisive clustering is a top-down approach: that is, all observations start in one cluster, and splits are performed recursively as one moves down the hierarchy. In general, the merges and splits are determined in a greedy manner. The results of hierarchical clustering are usually presented in a dendrogram. The hierarchical clustering problem has been studied for several decades in a wide range of fields including manufacturing, biotechnology, information technology (IT), logistics and transportation, financial, and postal industries. In the manufacturing sector, hierarchical clustering has been used to form manufacturing cells and processing batches. Vakharia and Wemmerlöv [1] investigated the performance of seven hierarchical agglomerative clustering techniques and eight dissimilarity measures in the context of cell formation in the cellular manufacturing system. Chen et al. [2] proposed a constrained agglomerative clustering algorithm for the single batch processing machine scheduling problem often encountered in semiconductor manufacturing and metal heat treatment. Hierarchical clustering is one of the most commonly used methods in biotechnology for classification. Cheng et al. [3] suggested hierarchical model-based clustering of DNA sequences by upgrading Bayesian model-based 2 clustering. Cameron et al. [4] proposed hierarchical clustering of gene expression patterns consistent with the lineage of differentiating excitatory neurons during early neocortical development. Saunders et al. [5] used Markov clustering and hierarchical clustering to classify protein families of rust pathogens and rank them according to their likelihood of being effectors. Barzinpour et al. [6] proposed a spectral approach to community detection, where the multiplex is mapped onto Euclidean Space (using the top few eigenvectors) and applied 𝑘-mean clustering. See Andreopoulos et al. [7] for a review of the clustering algorithms applied in bioinformatics. Clustering is one of the most important techniques for image segmentation and data analytics in the IT industry. Arifin and Asano [8] presented a histogram thresholding algorithm using hierarchical cluster analysis for image segmentation. Nunez-Iglesias et al. [9] proposed an active machine learning approach for performing hierarchical agglomerative clustering from superpixels to improve segmentation of 2D and 3D images. See Zaitoun and Aqel [10] for a survey of image segmentation techniques. In relation to data analytics, Bouguettaya et al. [11] proposed a set of agglomerative hierarchical clustering methods, and Costa et al. [12] proposed a hierarchical approach for clustering XML documents with multiple forms of structural components. Hierarchical clustering also has been successfully applied to the logistics and transportation sector. Özdamar and Demir [13] proposed a multilevel clustering algorithm that groups demand nodes into smaller clusters at each planning level for coordinating vehicle routing in large-scale postdisaster distribution and evacuation activities. Zhu and Guo [14] extended the traditional hierarchical clustering method by generalizing flows to different hierarchical levels to aggregate and map large taxi flow data in an urban area. The hierarchical clustering problem arises in the postal industry as well. Lim et al. [15] studied the three-level presorting loading problem which occurs in the commercial bulk mail service. They considered the problem as a three-level hierarchical clustering problem and proposed an optimal solution algorithm. For the financial sector application, Aghabozorgi and Teh [16] suggested a novel three-phase clustering model to categorize companies based on the similarity in the shape of their stock markets. See Murtagh and Contreras [17] for an extensive survey on the agglomerative hierarchical clustering algorithms. In this st (...truncated)