The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...

this process is repeated endlessly. Because the simplex method produces a sequence with monotonically improving objective values, the objective stays constant in a cycle, thus each pivot in the cycle must be degenerate. The possibility of cycling was recognized shortly after the invention of the simplex algorithm. Cycling examples were given by E.M.L. Beale [2] and by A.J. Hoffman [10]. Recently a scheme to construct cycling LO examples is presented in [9]. These examples made evident that extra techniques are needed to ensure finite termination of simplex methods. The first and widely used such tool is the lexicographic simplex rule. Other techniques, like the leastindex anti-cycling rules and more general recursive schemes were developed more recently. Lexicographic simplex methods. First we need to define an ordering, the so-called lexicographic ordering of vectors. Lexicograph

In this paper a new method is suggested for solving the problem in which the objective function is a linear fractional function, and where the constraint functions are in the form of linear inequalities. The proposed method is based mainly upon revised primal dual simplex algorithm (RPDSA).The algorithm can be combined with interior-point methods to move from an interior point to a basic optimal solution. The advantages of RPDSA algorithm are simplicity of implementation and less computational effort.

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