which is the inverse problem of

linear programming and can be solved by means of learning algorithms of neural network

Keywords

Linear programming, simplex method, initial feasible basis, elementary transformation of matrix.

Linear programming models (LPM) are commonly used in many industries to maximize or minimize some quantity (usually profits or costs) (Anderson et al.

in

linear programming. Any

linear programming software package such as Lindo would be capable of running this model.

Finally, in Appendix A, we demonstrate an integrality gap of D for a

linear programming formulation stronger than that used for parallel disk systems and, in Appendix B, that if no extra memory locations are allowed, then the integrality gap of our linear program can be arbitrarily large.

(Two appendices spell out the linear program in more detail: Appendix A contains a short review of linear programs, and Appendix B derives the

linear programming representation of the moral-hazard program.) Next, we compute the solution to a bank regulation example followed by some concluding comments.

Additional Key Words and Phrases: Clustering, collision detection,

linear programming, matrix searching, parametric searching, proximity problems, prune-and-search, randomized algorithms

Linear programming is a powerful mathematical technique that is "infrequently applied to smaller problems."(4) At one time the primary reason for this was the investment required in programming software.

Having arrived this far in developing a rationale for the application potential of

linear programming, the stage is now set for a call to lend active support towards implementing management science techniques in Pakistan.

The literature is also replete with articles on

linear programming approaches to portfolio selection and optimization which do take into consideration the individual preferences of investors (e.g., Lee & Lerro, 1973; Beazer, 1976; Mitchell & Rosebery, 1977; Courtney, 1979; Avery; Hodges & Schaefer, 1977).

The calculation of the above efficiency measures involves the use of

linear programming methods.

the constrained multiple objective

linear programming model) are quite complex, and we refer the reader to alternative sources for the complete methodological details.