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Optimization Using Linear Programming

A. J. Meitei and Veena Jain
Mercury Learning
Publication Date: 
Number of Pages: 
[Reviewed by
Brian Borchers
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Optimization Using Linear Programming is an undergraduate-level textbook on linear programming, the simplex method, duality theory, transportation and assignment problems, and two-person games.  Mathematical theory and computational methods for all of these problems were developed by the 1960's, but research in these areas is still ongoing.  Sadly, this textbook focuses on early approaches to these problems and misses more recent developments in the implementation of the simplex method as well as interior-point methods for linear programming.
The simplex method is presented here in the tableau format with elementary row operations used to update the tableau in each iteration of the simplex method.  In reading the book, I was struck by how many pages consist of simplex tableaus with very little explanatory text. In this format, the simplex method can seem like little more than an exercise in manipulating tables of numbers.
There is no discussion of the revised simplex method, sparse matrices, or techniques for updating basis factorizations, This has the advantage that the reader doesn't have to be very comfortable with algorithms for numerical linear algebra, but has the significant disadvantage of teaching the reader methods that are only appropriate for small problems rather than the much larger problems that occur in practice.
Throughout the book, the MS Excel solver is also used to solve the LP examples.  This is an easy way for the reader to check the answer, but if Excel has a built-in solver that can easily solve these problems, why should the student spend hours practicing techniques for solving the problems by hand?  Instead, why not focus more on the mathematical theory of linear programming, or on computational methods for large scale linear programming problems, or on modeling with linear programs?
A classic textbook that covers the simplex method and duality theory with more mathematical depth is Linear Programming by Vasek Chvatal. For a more complete and up-to-date treatment of simplex and interior-point methods for LP, see Robert Vanderbei's Linear Programming: Foundations and Extensions.  Model Building in Mathematical Programming by H. P. Williams is a very good textbook on formulating linear programming models.
Brian Borchers is a professor of mathematics at New Mexico Tech and the editor of MAA Reviews.
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