Stephanie Stahl

A Simulated Annealing Heuristic for a Bicriterion Partitioning Problem in Market Segmentation

K-means clustering procedures are frequently used to identify homogeneous market segments on the basis of a set of descriptor variables. In practice, however, market research analysts often desire both homogeneous market segments and good explanation of an exogenous response variable. Unfortunately, the relationship between these two objective criteria can be antagonistic, and it is often difficult to find clustering solutions that yield adequate levels for both criteria. The authors present a simulated annealing heuristic for solving bicriterion partitioning problems related to these objectives. A large computational study and an empirical demonstration reveal the effectiveness of the methodology. The authors also discuss limitations and extensions of the method.

Using Quadratic Assignment Methods to Generate Initial Permutations for Least-Squares Unidimensional Scaling of Symmetric Proximity Matrices

Combinatorial solution procedures for least-squares unidimensional scaling of symmetric proximity matrices frequently consist of two integrated processes: (a) the identification of a permutation of objects, and (b) the estimation of coordinate values on the continuum. These procedures typically require an initial permutation of objects. It is generally known that their final unidimensional scaling solutions are often very sensitive to these starting permutations, particularly when the number of objects is large (> 20). This paper demonstrates that, relative to random starting permutations, substantial improvements in final seriation quality and computational efficiency can be realized by using starting permutations obtained via solution to a quadratic assignment problem (QAP). Three methods-locally-optimal pairwise interchange (LOPI), simulated annealing (SA) and a hybrid (LOPI-SA)-were evaluated regarding their effectiveness and efficiency for solving the QAP. The results revealed that SA and LOPI-SA efficiently provided very good QAP solutions that subsequently led to good least-squares solutions.

An interactive multiobjective programming approach to combinatorial data analysis

Combinatorial optimization problems in the social and behavioral sciences are frequently associated with a variety of alternative objective criteria. Multiobjective programming is an operations research methodology that enables the quantitative analyst to investigate tradeoffs among relevant objective criteria. In this paper, we describe an interactive procedure for multiobjective asymmetric unidimensional seriation problems. This procedure uses a dynamic-programming algorithm to partially generate the efficient set of sequences for small to medium-sized problems, and a multioperation heuristic to estimate the efficient set for larger problems. The interactive multiobjective procedure is applied to an empirical data set from the psychometric literature. We conclude with a discussion of other potential areas of application in combinatorial data analysis.

Bicriterion seriation methods for skew-symmetric matrices.

The decomposition of an asymmetric proximity matrix into its symmetric and skew-symmetric components is a well-known principle in combinatorial data analysis. The seriation of the skew-symmetric component can emphasize information corresponding to the sign or absolute magnitude of the matrix elements, and the choice of objective criterion can have a profound impact on the ordering. In this research note, we propose a bicriterion approach for seriation of a skew-symmetric matrix incorporating both sign and magnitude information. Two numerical demonstrations reveal that the bicriterion procedure is an effective alternative to direct seriation of the skew-symmetric matrix, facilitating favourable trade-offs among sign and magnitude information.

Compact integer-programming models for extracting subsets of stimuli from confusion matrices

This paper presents an integer linear programming formulation for the problem of extracting a subset of stimuli from a confusion matrix. The objective is to select stimuli such that total confusion among the stimuli is minimized for a particular subset size. This formulation provides a drastic reduction in the number of variables and constraints relative to a previously proposed formulation for the same problem. An extension of the formulation is provided for a biobjective problem that considers both confusion and recognition in the objective function. Demonstrations using an empirical interletter confusion matrix from the psychological literature revealed that a commercial branch-and-bound integer programming code was always able to identify optimal solutions for both the single-objective and biobjective formulations within a matter of seconds. A further extension and demonstration of the model is provided for the extraction of multiple subsets of stimuli, wherein the objectives are to maximize similarity within subsets and minimize similarity between subsets.

An algorithm for extracting maximum cardinality subsets with perfect dominance or anti‐Robinson structures

A common criterion for seriation of asymmetric matrices is the maximization of the dominance index, which sums the elements above the main diagonal of a reordered matrix. Similarly, a popular seriation criterion for symmetric matrices is the maximization of an anti-Robinson gradient index, which is associated with the patterning of elements in the rows and columns of a reordered matrix. Although perfect dominance and perfect anti-Robinson structure are rarely achievable for empirical matrices, we can often identify a sizable subset of objects for which a perfect structure is realized. We present and demonstrate an algorithm for obtaining a maximum cardinality (i.e. the largest number of objects) subset of objects such that the seriation of the proximity matrix corresponding to the subset will have perfect structure. MATLAB implementations of the algorithm are available for dominance, anti-Robinson and strongly anti-Robinson structures.