The Most Representative Composite Rank Ordering
Rank-ordering of individuals or objects on multiple criteria has many important practical applications. A reasonably representative composite rank ordering of multi-attribute objects/individuals or multi-dimensional points is often obtained by the Principal Component Analysis, although much inferior but computationally convenient methods also are frequently used. However, such rank ordering - even the one based on the Principal Component Analysis - may not be optimal. This can be demonstrated by several numerical examples. To solve this problem, the Ordinal Principal Component Analysis was suggested some time back (Korhonen, 1984; Korhonen and Siljamaki, 1998). However, this approach cannot deal with various types of alternative schemes of rank ordering, mainly due to its dependence on the method of solution by the constrained integer programming. We propose an alternative method of solution (Mishra, 2009). A Fortran computer program to solve the problem has also been provided. This program uses Repulsive Particle Swarm optimizer. The same thing can be accomplished by the Differential Evolution optimizer also. Thus, it yields another program (source codes provided here). The suggested method is notably versatile and can take care of various schemes of rank ordering, norms and types or measures of correlation. The versatility of the method and its capability to obtain the most representative composite rank ordering of multi-attribute objects or multi-dimensional points have been demonstrated by several numerical examples. The overall ranking of objects by maximin correlation principle performs better if the composite rank scores are obtained by direct optimization with respect to the individual ranking scores.
- Korhonen, P. (1984) "Ordinal Principal Component Analysis", HSE Working Papers, Helsinki School of Economics, Helsinki, Finland.
- Korhonen, P. and Siljamaki, A. (1998) "Ordinal Principal Component Analysis. Theory and an Application", Computational Statistics & Data Analysis, 26(4): 411-424.
- Mishra, S.K. (2009) "The Most Representative Composite Rank Ordering of Multi-Attribute Objects by the Particle Swarm Optimization". Available at SSRN: Download.
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