James G. Morris

Convergence of the Weiszfeld Algorithm for Weber Problems Using a Generalized “Distance” Function

This paper considers a generalization of the single and multisource Weber problem for the case when the “distance” function is some power K of the usual Ip distance. Properties of the generalized problem are established, and an appropriate generalization of the Weiszfeld iterative approach is given. A convergence proof is supplied for an ϵ-approximation to the original problem, under certain restrictions on p and K.

The Central Role of Engagement in Online Communities

Online communities are new social structures dependent on modern information technology, and they face equally modern challenges. Although satisfied members regularly consume content, it is considerably harder to coax them to contribute new content and help recruit others because they face unprecedented social comparison and criticism. We propose that engagement-a concept only abstractly alluded to in information systems research-is the key to active participation in these unique sociotechnical environments. We constructed and tested a framework that demonstrates what engagement is, where it comes from, and how it powerfully explains both knowledge contribution and word of mouth. Our results show that members primarily contribute to and revisit an online community from a sense of engagement. Nonetheless, word of mouth is partly influenced by prior satisfaction. Therefore, engagement and satisfaction appear to be parallel mediating forces at work in online communities. Both mediators arise from a sense of communal identity and knowledge self-efficacy, but engagement also emerges from validation of self-identity. Nevertheless, we also found signs that the contributions of the most knowledgeable users are not purely from engagement, but also from a competing sense of self-efficacy. Our findings significantly contribute to the area of information systems by highlighting that engagement is a concrete phenomenon on its own, and it can be directly modeled and must be carefully managed.

Linear facility location — Solving extensions of the basic problem

Abstract The basic problem is to locate a linear facility to minimize the sume of weighted shortest Euclidean distances from demand points to the facility. We extend the analysis to locating a constrained linear facility, a radial facility, a linear facility where distances are rectangular and a linear facility under the minimax criterion. Each case is shown to admit a simple solution technique.

On estimating road distances by mathematical functions

Abstract Empirical results are presented that are in contrast to those reported earlier in this journal by Berens and Korling. Based on these results, a modelling philosophy is stressed and the practical impact of using empirically based distance-predicting functions is discussed.

Technical Note—Minisum I p Distance Location Problems Solved via a Perturbed Problem and Weiszfeld's Algorithm

The classical iterative algorithm originally proposed by Weiszfeld for solving minisum facility location problems is generalized to the case of Ip distances. A differentiate approximating function is used to replace the original nondifferentiable convex objective function. The approximating function is shown to be uniformly convergent to the original objective function as a smoothing constant approaches zero. Characterizations and convergence properties peculiar to the algorithm are discussed, and computational experience is presented. The straightforward extension to multifacility location problems is also developed.

Technical Note—Solving Constrained Multi-Facility Location Problems Involving l p Distances Using Convex Programming

A method is described for approximating nondifferentiable convex minimization problems occurring in location theory by differentiable problems. Distances are generalized to lp distances that include rectangular and Euclidean distances as special cases. Any number of linear and/or nonlinear constraints can be accommodated. The method utilizes a hyperbolic distance function that is uniformly convergent to each lp distance. Computational results are reported.

Fitting hyperplanes by minimizing orthogonal deviations

Hyperplanes withm + 1 parameters are fitted by minimizing the sum of weighted orthogonal deviations to a set ofN points. There is no inverse regression incompatibility. For unweighted orthogonall1-fits essentially the same number of points are on either side of an optimal hyperplane. The criterion function is neither convex, nor concave, nor even differentiable. The main result is that each orthogonallp-fit interpolates at leastm + 1 points, for 0 <p ≤ 1. This enables the combinatorial strategy of systematically trying all possible hyperplanes which interpolatem + 1 data points.

Optimal objective function approximation for separable convex quadratic programming

We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori.

Parallel and serial successive overrelaxation for multicommodity spatial price equilibrium problems

Computational experience is reported for various serial implementations of successive overrelaxation (SOR) applied to a linear multicommodity spatial price equilibrium problem posed as linear complementarity problem. Computational schemes that neglect the vast majority of the variables during most iterations are shown to be relatively efficient. A parallel implementation based on the same neglect is shown to exhibit encouraging average speedup over the single processor case. The SOR approach is shown empirically to converge for nonsymmetric problems. Dense network problems with up to 60 regions (each a potential supply or demand region) and 10 commodities, representing on the order of 36,000 variables, as well as sparse network problems with up to 140 regions, are typically solved in a few seconds of CPU time.

Objective Function Approximation: An Application to Spatial Price Equilibrium Models

This paper describes a method for selecting a grid size for the piece wise-linear approximation of a separable quadratic program. It describes and demonstrates application of the procedure to a spatial price equilibrium model. The method is based on an a priori bound relating the amount of objective function approximation error to the suboptimality induced in the solution of the actual optimization problem.