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Two-phase Method without Any artificial variable

Northeastern Mathematical Journal 2008年第5期24卷 395-398页

作者：梁平张旭利张相斌 College of lnformation Dalian University Dalian 116622 College of Mathematics Jilin University Changchun 130012 College of Ecnomics and Managemen Nanjing University of Posts and Telecommunications Nanjing 210046

A method is provided to achieve an initial basic feasible solution of a linear programming in this paper. This method dose not need introducing any artificial variable, but needs only solving an auxiliary linear programming. Compared with the traditional two-phase method, it has advantages of saving the memories and reducing the computational efforts.

关键词： initial basic feasible solution auxiliary linear programming two-phase method artificial variable

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Enhanced Short Term Load Forecasting Using artificial variables

Enhanced Short Term Load Forecasting Using Artificial Variab...

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International Conference on Smart Grids and Energy Systems (SGES)

作者： Mohit, F. Sheikh, A. Veermata Jijabai Technol Inst Control & Decis Res Ctr Elect Engn Dept Mumbai Maharashtra India

ISBN: (纸本)9781728185507

With the ongoing development in the field of smart meters, the electricity consumption data is readily available and can be forecasted as well as analyzed for better planning of grid such as load scheduling, demand-side management, etc. In this paper, machine learning algorithms are utilized to develop a hybrid model for forecasting electricity consumption for a shorter period. The development is made possible by combining the simple algorithms as children and one of the best suitable algorithm as a parent. The parent algorithm combines the output of all the children to generate the final forecast. This hybrid model is labeled as "Blended DKFD" (Decision tree, K-Nearest Neighbor, Feedforward neural network, and Deep feedforward neural network) model. This model not only retains the advantages of children algorithms but also eliminates their drawbacks. In addition to hybridization, the inclusion of intentionally processed variables called artificial variables elevates the accuracy of the prediction. The study presented in this article aims at simplification of the development of prediction models, such that any interested body can develop a model suitable for their application cheaply, which will be capable of high accuracy and reliability. The Blended DKFD model is tested on the IEEE 39-bus system for different datasets considering artificial variables, historical data, and environmental factors. Finally, from the simulation results, it has been proved that the proposed model exhibits a low standard deviation with high accuracy depending on the children's algorithms and various variables used to build prediction models.

关键词： artificial variable Feed-forward Neural Network Load Forecasting Machine Learning

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Initialization of the simplex algorithm: An artificial-free approach

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SIAM REVIEW 1997年第4期39卷 736-744页

作者： Arsham, H Univ Baltimore Baltimore MD 21201 USA

The simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point. pie present a new general purpose solution algorithm which obviates the use of artificial variables. The algorithm searches for a feasible segment of a boundary hyperplane (a face of feasible region or an intersection of several faces) by using rules similar to the ordinary simplex.

关键词： linear programming basic variable set artificial variable advanced basis

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An artificial-free simplex-type algorithm for general LP models

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MATHEMATICAL AND COMPUTER MODELLING 1997年第1期25卷 107-123页

作者： Arsham, H Information and Quantitative Sciences University of Baltimore Baltimore MD 21201-5779 U.S.A.

The simplex algorithm requires additional variables (artificial variables) for solving linear programs which lack feasibility at the origin point. Some students, however, particularly nonmathematics majors, have difficulty understanding the intuitive notion of artificial variables. A new general purpose solution algorithm obviates the use of artificial variables. The algorithm consists of two phases. Phase 1 searches for a feasible segment of the boundary hyper-plane (a face of feasible region or an intersection of several faces) by using rules similar to the ordinary simplex. Each successive iteration augments the basic variable set, BVS, by including another hyper-plane, until the BVS is full, which specifies a feasible vertex. In this phase, movements are on faces of the feasible region rather than from a vertex to a vertex. This phase terminates successfully (or indicates the infeasibility of the problem) with a finite number of iterations, which is at most equal to the number of constraints. The second phase uses exactly the ordinary simplex rules (if needed) to achieve optimality. This unification with the simplex method is achieved by augmenting the feasible BVS, which is always initially considered empty at the beginning of Phase 1. The algorithm working space is the space of the original (decision, slack and surplus) variables in the primal problem. It also provides a solution to the dual problem with useful information. Geometric interpretation of the strategic process with some illustrative numerical examples are also presented.

关键词： linear programming initialization of simplex method rules basic variable set dual problem artificial variable

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A computer implementation of the Push-and-Pull algorithm and its computational comparison with LP simplex method

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APPLIED MATHEMATICS AND COMPUTATION 2005年第1期170卷 36-63页

作者： Arsham, H Cimperman, G Damij, N Damij, T Grad, J Univ Baltimore MIS Div Baltimore MD 21201 USA Petrol Co Ljubljana 1000 Slovenia Univ Ljubljana Fac Econ Ljubljana 1000 Slovenia Univ Ljubljana Sch Publ Adm Ljubljana 1000 Slovenia

The simplex algorithm requires artificial variables for solving linear programs, which lack primal feasibility at the origin point. We present a new general-purpose solution algorithm, called Push-and-Pull, which obviates the use of artificial variables. The algorithm consists of preliminaries for setting up the initialization followed by two main phases. The Push Phase develops a basic variable set (BVS) which may or may not be feasible. Unlike simplex and dual simplex, this approach starts with an incomplete BVS initially, and then variables are brought into the basis one by one. If the BVS is complete, but the optimality condition is not satisfied, then Push Phase pushes until this condition is satisfied, using the rules similar to the ordinary simplex. Since the proposed strategic solution process pushes towards ail optimal solution, it may generate all infeasible BVS. The Pull Phase pulls the solution back to feasibility using pivoting rules similar to the dual simplex method. All phases use the usual Gauss pivoting row operation and it is shown to terminate successfully or indicates unboundedness or infeasibility of the problem. A computer implementation, which enables the user to run either Push-and-Pull or ordinary simplex algorithms, is provided. The fully coded version of the algorithm is available from the authors upon request. A comparison analysis to test the efficiency of Push-and-Pull algorithm comparing to ordinary simplex is accomplished. Illustrative numerical examples are also presented. (c) 2004 Elsevier Inc. All rights reserved.

关键词： linear programming basic variable set artificial variable advanced basis simplex tableau reduction

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Improvement and its computer implementation of an artificial-free simplex-type algorithm by Ars ham

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APPLIED MATHEMATICS AND COMPUTATION 2015年 263卷 410-415页

作者： Gao, Pei-wang Minjiang Univ Dept Math Fuzhou 350108 Peoples R China

Arsham ever described a new phase 1 simplex-type algorithm which allegedly obviates the use of artificial variables. Soon after, Enge and Huhn gave a counterexample, in which Arsham's algorithm declares the infeasibility of a feasible problem. For this reason, this paper proposes an improvement of Arsham's algorithm to make it work well, which is searching for the nonbasic variables into the basic variable set (BVS) column by column in only one pivot sequence from beginning to end in order to decrease computational time spent by many repeated searching sequences after each iteration. Next, when the BVS is complete, the phase 1 algorithm ends with a feasible basic solution;otherwise, two variants are presented to pivot the artificial variables out of the basis. The idea of variant 1 is making all artificial variables in basis minimized in turn while maintaining the BVS feasible in the pivoting process. In variant 2, the objective is to minimize the sum of all artificial variables in basis while keeping basic variables (including artificial) feasible. Finally, a computer implementation is accomplished to test the efficiency of our improvement comparing to the ordinary simplex algorithm on some standard test instances and randomly generated problems. The computational results show that our improved algorithms averagely spends much less executive time at each iteration than the ordinary simplex algorithm on sparse linear programming problems, and variant 2 is also competitive on dense linear programming problems. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Linear programming Simplex method Phase 1 artificial variable Basic variable set (BVS)

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A COMBINED PHASE-I PHASE-II SCALED POTENTIAL ALGORITHM FOR LINEAR-PROGRAMMING

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MATHEMATICAL PROGRAMMING 1991年第3期52卷 429-439页

作者： ANSTREICHER, KM 1. Department of Management Sciences University of Iowa 52242 Iowa City IA USA

We develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program, without the addition of any "M" terms. The method, and its lower-bounding procedure, are particularly simple compared with previous interior algorithms not requiring feasibility.

关键词： LINEAR PROGRAMMING POTENTIAL FUNCTION PHASE-I PHASE-II artificial variable

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OPTIMAL LINEAR-PHASE DIGITAL-FILTER DESIGN BY ONE-PHASE LINEAR-PROGRAMMING

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS 1990年第4期37卷 554-558页

作者： JOHNSON, AT Department of Electrical Engineering School of Engineering Widener University Chester PA USA

An improved linear programming algorithm for the design of optimal linear-phase FIR filters with extra time-domain or frequency-domain constraints is presented. After stating the problem as involving the minimization of a linear function of filter parameters with linear constraints, a method is given for choosing an initial basic feasible solution of the dual problem that does not require the introduction of artificial variables. This significantly reduces the number of iterations of the revised simplex algorithm needed to reach the optimal solution.

关键词： linear phase optimal solution linear function filter design linear programming Algorithm design Optimal filter parameters iterative methods artificial variable

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ON INTERIOR ALGORITHMS FOR LINEAR-PROGRAMMING WITH NO REGULARITY ASSUMPTIONS

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OPERATIONS RESEARCH LETTERS 1992年第4期11卷 209-212页

作者： ANSTREICHER, KM Department of Management Sciences University of Iowa Iowa City IA USA

The linear programming algorithm of Karmarkar (1984), and all interior methods subsequently devised, require a regularity assumption that the primal and/or dual problem possess a nonempty interior. In this note we devise a polynomial-time interior algorithm which will directly 'process' and linear program, with no regularity assumptions whatsoever, without the addition of any 'M' terms. Our method is based on the application of a combined phase I-phase II algorithm to a combined primal-dual problem.

关键词： LINEAR PROGRAMMING INTERIOR ALGORITHM PHASE-I PHASE-II artificial variable

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Finding an Initial Basic Feasible Solution for DEA Models with an Application on Bank Industry

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COMPUTATIONAL ECONOMICS 2015年第2期45卷 323-336页

作者： Toloo, Mehdi Masoumzadeh, Atefeh Barat, Mona Tech Univ Ostrava Fac Econ Dept Business Adm Ostrava Czech Republic Islamic Azad Univ Cent Tehran Branch Dept Math Tehran Iran Islamic Azad Univ Mahshahr Branch Dept Math Mahshahr Iran

Nowadays, algorithms and computer programs, which are going to speed up, short time to run and less memory to occupy have special importance. Toward these ends, researchers have always regarded suitable strategies and algorithms with the least computations. Since linear programming (LP) has been introduced, interest in it spreads rapidly among scientists. To solve an LP, the simplex method has been developed and since then many researchers have contributed to the extension and progression of LP and obviously simplex method. A vast literature has been grown out of this original method in mathematical theory, new algorithms, and applied nature. Solving an LP via simplex method needs an initial basic feasible solution (IBFS), but in many situations such a solution is not readily available so artificial variables will be resorted. These artificial variables must be dropped to zero, if possible. There are two main methods that can be used to eliminate the artificial variables: two-phase method and Big-M method. Data envelopment analysis (DEA) applies individual LP for evaluating performance of decision making units, consequently, to solve these LPs an IBFS must be on hand. The main contribution of this paper is to introduce a closed form of IBFS for conventional DEA models, which helps us not to deal with artificial variables directly. We apply the proposed form to a real-data set to illustrate the applicability of the new approach. The results of this study indicate that using the closed form of IBFS can reduce at least 50 % of the whole computations.

关键词： Data envelopment analysis Initial basic feasible solution artificial variable Two-phase method

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