Canonical analysis of two convex polyhedral cones and applications

Canonical analysis of two convex polyhedral cones consists in looking for two vectors (one in each cone) whose square cosine is a maximum. This paper presents new results about the properties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of scalings of an ordinal variable is a convex polyhedral cone, which thus plays an important role in optimal scaling methods for the analysis of ordinal data. Monotone analysis of variance, and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace. Optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given. We discuss these three situations and illustrate them by examples.     

Experiments comparing qualitative approaches to rank ordering of multiattribute alternatives

ZAPROS, a method to support rank ordering tasks using ordinal input from decision makers, is discussed and compared with a preference cone technique and the analytic hierarchy process (AHP). It provides a means to identify inconsistencies in ordinal decision tasks, yielding verification and explanation of results for partial ordering of a large set of alternatives. The results indicate that ZAPROS provides no less accuracy in task solution, while having some advantages from a behavioural point of view. Comparative analysis of the effectiveness of the methods under consideration in accordance with differences in task characteristics is carried out.     

The Love equation for the normal loading of a rigid cone on an elastic half-space: no need for a modification

A new schematic diagram is presented which shows the displacement paths of surface particles within the contact circle when a rigid cone is loaded normally on an elastic half-space. With the help of this diagram and the information available in the literature, it is argued here that a modification made to the Love equation in 1999 is unnecessary and incorrect. It is also shown that the modification results in an incorrect expression for the surface radial displacement outside the contact circle. Therefore, it is suggested that the modification be dropped forthwith.     

A new solution to the problem of finding all numerical solutions to ordered metric structures

A new algorithm is used to test and describe the set of all possible solutions for any linear model of an empirical ordering derived from techniques such as additive conjoint measurement, unfolding theory, general Fechnerian scaling and ordinal multiple regression. The algorithm is computationally faster and numerically superior to previous algorithms.This research was supported in part by NIGMS grant GM-01231 to the University of Michigan. Authors' names are in alphabetic order.     

Matching, undermatching, and overmatching in studies of choice

Almost all of 103 sets of data from 23 different studies of choice conformed closely to the equation: log (B(1)/B(2)) = a log (r(1)/r(2)) + log b, where B(1) and B(2) are either numbers of responses or times spent at Alternatives 1 and 2, r(1) and r(2) are the rates of reinforcement obtained from Alternatives 1 and 2, and a and b are empirical constants. Although the matching relation requires the slope a to equal 1.0, the best-fitting values of a frequently deviated from this. For B(1) and B(2) measured as numbers of responses, a tended to fall short of 1.0 (undermatching). For B(1) and B(2) measured as times, a fell to both sides of 1.0, with the largest mode at about 1.0. Those experiments that produced values of a for both responses and time revealed only a rough correspondence between the two values; a was often noticeably larger for time. Statistical techniques for assessing significance of a deviation of a from 1.0 suggested that values of a between .90 and 1.11 can be considered good approximations to matching. Of the two experimenters who contributed the most data, one generally found undermatching, while the other generally found matching. The difference in results probably arises from differences in procedure. The procedural variations that lead to undermatching appear to be those that produce (a) asymmetrical pausing that favors the poorer alternative; (b) systematic temporal variation in preference that favors the poorer alternative; and (c) patterns of responding that involve changing over between alternatives or brief bouts at the alternatives.     

Relative importance of criteria: a quantitative approach

A mathematical definition for the assertion ‘a group of criteria A is more important than a group of criteria B’ is introduced. It is shown how quantitative information on relative importance of criteria allows us to obtain a more precise upper estimate for a set of all non-dominated solutions than the well-known Pareto set. © 1997 John Wiley & Sons, Ltd.