A generalization of the MDS method by mixed integer linear and nonlinear mathematical models
The Multi-Dimensional Scaling (MDS) method is used in statistics to detect hidden interrelations among multi-dimensional data and it has a wide range of applications. The method’s input is a matrix that describes the similarity/dissimilarity among objects of unknown dimension. The objects are generally reconstructed as points of a lower dimensional space to reveal the geometric configuration of the objects. The original MDS method uses Euclidean distance, for measuring both the distance of the reconstructed points and the bias of the reconstructed distances from the original similarity values. In this paper, these distances are distinguished, and distances other than Euclidean are also used, generalizing the MDS method. Two different distances may be used for the two different purposes. Therefore the instances of the generalized MDS model are denoted as model, where the first distance is the type of distance of the reconstructed points and the second one measures the bias of the reconstructed distances and the similarity values. In the case of and distances mixed-integer programming models are provided. The computational experiences show that the generalized model can catch the key properties of the original configuration, if any exist.
Keywords: Multivariate Analysis; Multi-Dimensional Scaling; Optimization; Mixed Integer Linear Programming; Statistics.
P. M. Hahn, J. Krarup, A hospital facility layout problem ﬁnally solved. Journal of Intelligent Manufacturing 12 (2001) 487–496. http://dx.doi.org/10.1023/A:1012252420779.
J. T. Machado, G. M. Duarte, F. B. Duarte, Identifying Economic Periods and Crisis with the Multidimensional Scaling. Nonlinear Dynamics 4 (2010) 611-622.
M. Smallman-Raynor, A. D. Cliff, Epidemiological spaces: the use of multidimensional scaling to identify cholera diffusion processes in wake of the Philippines insurrection, 1899–1902. Trans Inst Br Geogr 26 (2001) 288-305. http://dx.doi.org/10.1111/1475-5661.00023.
M. Syrquin, the Application of Multidimensional Scaling to the Study of Economic Development. The Quarterly Journal of Economics 92(4) (1978) 621-639. http://dx.doi.org/10.2307/1883179.
N. Jaworska, A. Chupetlovska‐Anastasova, a Review of Multidimensional Scaling (MDS) and its Utility in Various Psychological Domains. Tutorials in Quantitative Methods for Psychology 5(1) (2009) 1-10.
S. Niroomand, S. Takács, B. Vizvári, To lay out or not to lay out? Annals of Operations Research 191 (2011) 183-192. http://dx.doi.org/10.1007/s10479-011-1005-1.
S. Niroomand, B. Vizvari, Exact mathematical formulations and metaheuristic algorithms for production cost minimization: a case study of the cable industry. International Transactions in Operational Research (2014). DOI: 10.1111/itor.12096. http://dx.doi.org/10.1111/itor.12096.