Mathematical Programming-based Approach Research Articles

Using mathematical programming to compute singlular multivariate normal probablities

This paper describes two new, mathematical programming-based approaches for evaluating general, one- and two-sidedp-variate normal probabilities where the variance-covariance matrix (of arbitrary structure) is singular with rankr(r<pand r and p can be of unlimited dimensions. In both cases, principal components are used to transform the original, ill-definedp-dimensional integral into a well-definedrdimensional integral over a convex polyhedron. The first algorithm that is presented uses linear programming coupled with a Gauss-Legendre quadrature scheme to compute this integral, while the second algorithm uses multi-parametric programming techniques in order to significantly reduce the number of optimization problems that need to be solved. The application of the algorithms is demonstrated and aspects of computational performance are discussed through a number of examples, ranging from a practical problem that arises in chemical engineering to larger, numerical examples.

A genetic algorithm for a single hoist scheduling in the printed-circuit-board electroplating line

In this paper, the problem of determining cyclic schedules for a material handling hoist in the printed-circuit-board(PCB) electroplating line is considered. The objective of this research is to determine an optimal simple-cycle schedule of the hoist which in turn maximizes the line throughput rate. Previous approaches to the cyclic hoist scheduling problem are all mathematical programming-based approaches to develop cyclic schedules(Mixed Integer Programming, Linear Programming based Branch and Bound, Branch and Bound Search Method and so on). In this paper, a genetic algorithm-based approach for a single hoist scheduling in the PCB electroplating line is described. Through an experiment for the well known example data, the proposed algorithm is shown to be more efficient than the previous mathematical programming-based algorithm.

Designing environmentally safe refrigerants using mathematical programming

Computer-aided molecular design is a strategy in which a set of structural groups are systematically combined to form molecules with desired properties. In this paper, a mathematical programming-based approach to computer-aided molecular design is presented. Using a set of structural groups, the problem is formulated as a mixed integer nonlinear program in which discrete variables represent the number of each type of structural groups present in the candidate compound. The augmented-penalty/outer-approximation algorithm is used to solve the MINLP to obtain compound(s) with an optimum value of an appropriate performance index such that molecular structural constraints, physical property constraints and process design limitations are met. With the current renewed interest in the environment, the suggested approach is applied to refrigerant design with an environmental constraint. The results indicate the viability of this approach.

On the classification gap in mathematical programming-based approaches to the discriminant problem

This article proposes a mathematical-programming-based approach to solve the classification problem in discriminant analysis which explicitly considers the classification gap. The procedure consists of two distinct phases and initially treats the classification gap as a fuzzy set in which the classification rule is not yet established. The nature of the classification gap is examined and a variety of methods are discussed which can be applied to identify the most appropriate classification rule over the fuzzy set. The proposed methodology has several potential advantages. First, it offers a more refined approach to the classification problem, facilitating careful analysis of the fuzzy region where the classification decision may not be obvious. Secondly, the two-phase approach enables the analysis of larger data sets when using computer-intensive procedures such as mixed-integer programming. Finally, because of the restricted choice of separating hyperplanes in phase 2, the approach appears to be more robust than other classification techniques with respect to outlier-contaminated data conditions. The robustness issue and computational advantage of our proposed methodology are illustrated using a limited simulation experiment.