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AbstractAlthough cycling in the simplex method has long been known, a number of theoretical questions concerning cycling have not been fully answered. One of these, stated in [3], is to find the smallest example of cycling, and Beale's example with three equations and seven variables is conjectured to be the smallest one. The exact bounds on dimensions of cycling examples are established in this paper. We show that Beale's example is the smallest one which cycles at a non‐optimal solution, that a smaller one can cycle at the optimum, and that, in general (including the completely degenerate case), a cycling example must have at least two equations, at least six variables, and at least three non‐basic variables. Examples and geometries are given for the extreme cases, showing that the bounds are sharp.

The productivity and quality of working life of operations within a
firm with multiple operating units can be improved by the diffusion of
incremental improvements from one unit to another. The opportunities and
problems experienced by one large multi‐unit firm in diffusing employee
innovations between units are examined. It was found that incremental or
Minor Technical Improvements (MTIs) do diffuse between units. Improved
communication and increased under standing of the innovation process by
management may increase the amount of MTI diffusion. The costs and
benefits of MTI type innovation and diffusion are explored.
Opportunities for more research into the diffusion of MTIs in other
multi‐unit firms are discussed.

ABSTRACTTorrance [4] has proposed a "new" approach to finding an initial solution of a linear programming problem for use in conjunction with the dual simplex algorithm. The purpose of this note is to comment on two aspects of that paper. Firstly, Torrance's method is not new at all, but was proposed in 1958 by Wagner. Secondly, the method can be implemented much more efficiently than Torrance suggests.

AbstractIn this paper we describe the principle of external pivoting, a variant of the primal simplex algorithm. Its validity is proved in general and various strategies are designed which are based on this principle. The average convergence behavior for these strategies is then tested in a series of randomly generated problems. It is found that at least one of the strategies compares favorably with the regular simplex method.

A variety of pivot column selection rules based upon the gradient criteria
(including the steepest edge) have been explored to improve the efficiency of the primal
simplex method. Simplex-like algorithms have been proposed imbedding the gradient
direction (GD) which includes all variables whose increase or decrease leads to an improvement
in the objective function. Recently a frame work has been developed in the
simplex method to incorporate the reduced-gradient direction (RGD) consisting of only
variables whose increase leads to an improvement in the objective function. In this paper,
the results are extended to embed GD in the simplex method based on the concept
of combining directions. Also mathematical properties related to combining directions
as well as deleting a variable from all basic directions are presented.

AbstractThis note describes the construction of an example in which the usual iterative method for linear programming may fail unless special techniques for overcoming degeneracy are used.

Within the policy adoption process of Forest Management Unit (FMU) concept, there has been disagreement of stakeholders on FMUs concept. This disagreement is caused by the exchange of knowledge, information, and perception among stakeholders involved. The results of these interactions could speed up, slow down, and prevent the adoption process of FMU policy. The study objective was analyzing process of knowledge diffusion of FMUs development policy and stakeholders interaction in PFMU Batutegi and PFMU Kotaagung Utara, Indonesia. Adoption process was analyzed by the logical diffusion technique based on knowledge time of FMUs concept received and its interaction space. Social interaction among stakeholders was analyzed using method developed by International Development Studies analysis, i.e. interaction among discourse/narrative, actors/networks and politics/interests. The results showed that knowledge diffusion of FMUs concept in both PFMU tends to cascade diffusion. Factors was affecting of it process were network, role of opinion leaders, willingness to know, and understand on FMUs concept. Indicative strategy is needed as anticipating and overcoming an obstacle in its internalization process, i.e. harmonization of legislative and executive relationship, building an opinion the importance of FMU, and optimalizing network for bureaucratic problems.