A set of mathematical rules for solving a particular class of problem or model is known as an *algorithm*. We are interested in algorithms for solving linear programming (LP), separable programming and integer programming IP models. An algorithm can be programmed into a set of computer routines for solving the corresponding type of model assuming the model is presented to the computer in a specified format. For algorithms that are used frequently, it turns out to be worth writing very sophisticated and efficient computer programmes for use with many different models. Such programmes usually consist of a number of algorithms collected together as a ‘*package*’ of computer routines. Many such package programmes are available commercially for solving mathematical programming models. They usually contain algorithms for solving LP models, separable programming models and IP models. These packages are written by computer manufacturers, consultancy firms and software houses. They are frequently very sophisticated and represent many person-years of programming effort. When a mathematical programming model is built, it is usually worth making use of an existing package to solve it rather than getting diverted onto the task of programming the computer to solve the model oneself.

The algorithms that are almost invariably used in commercially available packages are (i) the revised simplex algorithm for LP models; (ii) ...

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