Field Of Linear Programming Research Articles

THE MAXIMUM RANGE COLUMN METHOD – GOING BEYOND THE TRADITIONAL INITIAL BASIC FEASIBLE SOLUTION METHODS FOR THE TRANSPORTATION PROBLEMS

The transportation problems (TPs) are a fundamental case-study topic in operations research, particularly in the field of linear programming (LP). The TPs are solved in full resolution by using two types of methods: initial basic feasible solution (IBFS) and optimal methods. In this paper, we suggest a novel IBFS method for enhanced reduction in the transportation cost associated with the TPs. The new method searches for the range in columns of the transportation table only, and selects the maximum range to carry out allocations, and is therefore referred to as the maximum range column method (MRCM). The performance of the proposed MRCM has been compared against three traditional methods: North-West-Corner (NWCM), Least cost (LCM) and Vogel’s approximation (VAM) on a comprehensive database of 140 transportation problems from the literature. The optimal solutions of the 140 problems obtained by using the TORA software with the modified distribution (MODI) method have been taken as reference from a previous benchmark study. The IBFSs obtained by the proposed method against NWCM, LCM and VAM are mostly optimal, and in some cases closer to the optimal solutions as compared to the other methods. Exhaustive performance has been discussed based on absolute and relative error distributions, and percentage optimality and nonoptimality for the benchmark problems. It is demonstrated that the proposed MRCM is a far better IBFS method for efficiently solving the TPs as compared to the other discussed methods, and can be promoted in place of the traditional methods based on its performance.

Dynamical Complex System. Multi-Objective Nonlinear Control System by Fuzzy Weighting Factor Method based on Universal Learning Network.

Control probrems on the large-scale complicated systems such as optimal distributed control and optimal hierarchical control have been extensively studied in the 1960's. On the other hand, recently, multi-objective optimization problems have been studied in the field of Linear Programming and Non Linear Programming problems. In this paper, a new control methodology called fuzzy weighting factor method is presented that automatically determines the weighting factors of the criterion function of the large-scale multi-objective non linear systems which are made up of plural subsystems with their own criterion functions. Hitherto, weighting factor method, e constraint method and fuzzy Min Max method etc. have been presented as the method of optimizing the multiobjective systems. Most of them are applied to static optimization probrems, and they have the problem to be solved such that as the methods are converted to linear programming problems, the membership functions of the plural criterion functions should be approximated linearly and the methods need a fairly amount of computations. In this paper, the controller's parameter variables in the dynamic multi-objective non-linear control systems are determined by using Universal Learning Network (ULN) and a new fuzzy weighting factor method. And the new method can solve the above mentioned problems and is also different from the usual fuzzy Min Max method in the point that the new fuzzy weighting factor method can arbitrarily adopt the membership functions. The Universal Learning Network is a super set of all kinds of neural network paradigms with supervised learning capability and in this paper both control objects and its controllers of the large-scale complicated systems are represented by ULN. Finally, characteristics of the fuzzy weighting factor method are studied by simulations on a nonlinear tank network.

Chaotic solutions in dynamic linear programming

In the field of linear programming, the fact that the optimal solution may follow chaotic non-linear dynamics has not been demonstrated. In this study, we demonstrate that chaotic dynamics may emerge as a solution to a dynamic linear programming problem with an infinite time horizon.

Probabilistic analysis of optimization algorithms-some aspects from a practical point of view

In this paper the utility and the difficulties of probabilistic analysis for optimization algorithms are discussed. Such an analysis is expected to deliver valuable criteria-better than the worst-case complexity-for the efficiency of an algorithm in practice. The author has done much work of that kind in the field of linear programming. Based on that experience he gives some insight into the general principles for such an approach. He reports on some typical and representative attempts to analyze algorithms, resp. problems, of linear and combinatorial optimization. For each case he describes the problem, the stochastic model under consideration, the algorithm, the results, and tries to give a brief idea of the way these results could be obtained. He concludes with a discussion of some drawbacks and difficulties in that field of research. Among these are the strong sensibility with respect to the chosen model, the restriction of results to the asymptotic case, the restriction to somehow inefficient algorithms, etc. These points are the reasons why probabilistic analysis is of limited value for practice today. On the other hand, they show which principal problems should be attacked in the future to obtain the desired utility.

Story problems and mathematical modelling: the effectiveness of the cognitive method

The aim of this paper is twofold: to show the necessity of teaching story problems in secondary school as preparation for mathematical modelling; and to call attention to the cognitive method: a method for transforming verbally‐given problems into mathematical form, either during the teaching of story problems in school, or while mathematical modelling in science, business, economics or industry is being carried out. The cognitive method concerns problems which lead to equations, inequalities or functions or their systems; so it concerns a variety of fields, including linear programming. The examples in this paper are in the field of linear programming.

Network Synthesis Through Hybrid Matrices

Abstract : It is shown that the hybrid parameter matrix of a passive n-port network is a positive real matrix function. The hybrid, chain and immittance parameter matrices all have the same degree. The concept of combinatorial equivalence is extended from the linear programming field to network synthesis. An n-port extension of the Brune synthesis procedure is given. A novel concept of degree enumeration is given. (Author)

On the Translocation of Masses

The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Académie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7–8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.—A. Charnes, Northwestern Technological Institute and The Transportation Center.

Recent Advances in Linear Programming

As interest grows rapidly in industry on the potentialities of mathematical programming techniques, it appears worthwhile to have a paper devoted to some of the more promising developments which may speed up the transition from interest to use. Three topics have been selected (in three sections that follow) which have recently come into prominence: uncertainty, combinatorial problems, and large scale systems. The reader will find in the course of their discussions that a survey—though perhaps not a systematic survey—has been made of current techniques in the linear programming field.