Academic journal article Journal of Management Information and Decision Sciences

Lexicographic Goal Programming Approaches to the Three-Group Classification Problem

Academic journal article Journal of Management Information and Decision Sciences

Lexicographic Goal Programming Approaches to the Three-Group Classification Problem

Article excerpt

INTRODUCTION

Over the last thirty years there has been considerable research interest in mathematical programming approaches to the statistical classification problem. Various simulation studies (Lee, Gallagher & Patterson, 2003; Bal, Orkcu & Celebioglu, 2006; Sueyoshi, 2006) have demonstrated that such mathematical programming approaches can outperform the classical statistical procedures for classification, i.e. the linear discriminant function (Fisher, 1936) and the quadratic discriminant function (Smith, 1947), when the conditions for optimality of these parametric methods are violated. These conditions for optimality include multivariate normality and assumptions about the covariance structures.

Mixed-integer programming models for the three-group classification problem (Gehrlein, 1986; Gochet, Stam, Srinivasan & Chen, 1997; Loucopoulos & Pavur, 1997) directly minimize the number of misclassifications in the training sample. It has been shown that mathematical programming models for the three-group classification problem outperform the standard parametric classification procedures (Fisher's linear discriminant function and Smith's quadratic discriminant function) for a variety of data configurations (Loucopoulos & Pavur, 1997b; Bal & Orkcu, 2010). It should be noted that depending on group separation and the number of observations, a mixed-integer programming model may have alternate solutions in the training sample. The choice of the alternate solution used for generating the discriminant function can have a significant effect on the holdout sample classificatory performance of the model. In this paper, various secondary goals are investigated for the three-group classification problem, with the goal of minimizing the number of misclassifications in the training sample being assigned preemptive priority.

MAXIMIZATION OF MAXIMUM DEVIATION BETWEEN PROJECTED GROUP

MEANS AS SECONDARY GOAL

The MIP3G model (Loucopoulos & Pavur, 1997) is a mixed-integer programming model specifically for the three-group classification problem. It assigns a weight [a.sub.k] to each attribute variable [X.sub.k] (k = 1, 2, ..., p) and thus generates a discriminant score [a.sub.0] + [p.summation over (k=1)] [a.sub.k][X.sup.(i).sub.k] for each observation i (i = 1, 2, ..., n). Such score is projected onto a line which is divided into three intervals, one for each group, with a gap of width e between adjacent intervals for enhanced group separation. An observation is correctly classified if its discriminant score falls in the interval assigned to its group; otherwise it is misclassified.

Notation:

[a.sub.k] is the weight assigned to variable [X.sub.k]

[X.sup.(i).sub.k] is the value of variable k observation i

[a.sub.0] is a locational adjustment to the discriminant function

e is the width of the middle interval

e is the width of the gap between adjacent intervals

[M.sub.1] is the maximum deviation of the discriminant score of a misclassified observation from the nearest endpoint of the interval assigned to its group

[M.sub.2] is the maximum deviation of the discriminant score of a correctly classified observation, that belongs to either the leftmost or rightmost group, from -e or e+ e, respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Formulation:

min [n.summation over (i=1)] [I.sub.i]

s.t.

[a.sub.0] + [p.summation over (k=1)] [a.sub.k] [X.sup.(i).sub.k] - [M.sub.1][I.sub.i] - (e + e') [K.sub.2] + ([M.sub.2] + e) [K.sub.3] [less than or equal to] [M.sub.2] [for all]i [member of] [G.sub.1] (1)

[a.sub.0] + [p.summation over (k=1)] [a.sub.k] [X.sup.(i).sub.k] - [M.sub.1][I.sub.i] - ([M.sub.2] + e) [K.sub.2] + (e + e') [K.sub.3] [greater than or equal to] e' - [M. …

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