An Artificial-Free Simplex-Type Algorithm for General LP Models (original) (raw)
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A TABULAR SIMPLEX-TYPE ALGORITHM AS A TEACHING AID FOR GENERAL LP MODELS
Mathematical and Computer Modelling , 1989
The simplex algorithm requires additional variables (artificial variables) for solving linear programs which lack feasibility at the origin point. However, some students, particularly non-mathematics 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 I 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 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 infeasibility of the problem) with a finite number of iterations, which is 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 empty in phase 1. The algorithm working space is the space of the original (decision, slack, and surplus) variables with a geometric interpretation of its strategic process.
Applied Mathematics and Computation , 2005
"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 an optimal solution, it may generate an 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 imple-mentation, 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. The software is available at: http://home.ubalt.edu/ntsbarsh/Research/LpSolvers.htm For teaching purposes you may try: A tabular simplex-type algorithm as a teaching aid for general LP models H Arsham Mathematical and Computer Modelling 12(8):1051-1056, 1989 Available at: http://home.ubalt.edu/ntsbarsh/Push\_pull\_original.pdf
A Computer Technique for Solving LP Problems with Bounded Variables
Dhaka University Journal of Science, 2012
Linear Programming problem (LPP)s with upper bounded variables can be solved using the Bounded Simplex method (BSM),without the explicit consideration of the upper bounded constraints. The upper bounded constraints are considered implicitly in this method which reduced the size of the basis matrix significantly. In this paper, we have developed MATHEMATICA codes for solving such problems. A complete algorithm of the program with the help of a numerical example has been provided. Finally a comparison with the built-in code has been made for showing the efficiency of the developed code.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11487 Dhaka Univ. J. Sci. 60(2): 163-168, 2012 (July)
Practical application of simplex method for solving linear programming problems
BALKAN JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS, 2018
In this paper we consider application of linear programming in solving optimization problems with constraints. We used the simplex method for finding a maximum of an objective function. This method is applied to a real example. We used the “linprog” function in MatLab for problem solving. We have shown, how to apply simplex method on a real world problem, and to solve it using linear programming. Finally we investigate the complexity of the method via variation of the computer time versus the number of control variables.
A Streamlined Artificial Variable Free Version of Simplex Method
PLOS ONE, 2015
This paper proposes a streamlined form of simplex method which provides some great benefits over traditional simplex method. For instance, it does not need any kind of artificial variables or artificial constraints; it could start with any feasible or infeasible basis of an LP. This method follows the same pivoting sequence as of simplex phase 1 without showing any explicit description of artificial variables which also makes it space efficient. Later in this paper, a dual version of the new method has also been presented which provides a way to easily implement the phase 1 of traditional dual simplex method. For a problem having an initial basis which is both primal and dual infeasible, our methods provide full freedom to the user, that whether to start with primal artificial free version or dual artificial free version without making any reformulation to the LP structure. Last but not the least, it provides a teaching aid for the teachers who want to teach feasibility achievement as a separate topic before teaching optimality achievement.
Implementation and Evaluation of Primal and Dual Simplex Methods
https://orcid.org/0009-0007-2904-0443, 2025
This research paper investigates advanced optimization techniques, focusing on the implementation and performance analysis of primal and dual simplex methods for solving linear programming (LP) problems. It provides a comparative evaluation of five distinct pivot-selection strategies: Non-Basic Gradient, Bland's method, Least-Recently Considered (LRC), Greatest-Increment, and Steepest-Edge methods. Each strategy has unique characteristics that affect computational efficiency and convergence speed, making them essential for practitioners in various optimization contexts. Found on pages: [ 12 ] Methodology and Environment The study employs the LPBench environment—a specialized platform designed for precise evaluation of linear programming algorithms. The analysis includes a standardized test suite comprising 500 LP problems with varying complexities, from small-scale instances (50-100 variables) to large-scale scenarios (over 1,000 variables). These problems encompass both randomly generated cases and real-world applications across domains like transportation, resource allocation, and production planning. Performance metrics are rigorously defined to assess pivot iteration counts, CPU time, memory usage, and numerical stability across different problem configurations. Found on pages: [ 2543 ] Findings and Comparisons Findings reveal notable trade-offs between iteration counts and computational complexity associated with each pivot-selection method. While Bland's method operates efficiently at O(n) per iteration—requiring 2-3 times more iterations to reach convergence compared to alternatives—the Non-Basic Gradient and LRC methods show comparable iteration counts. Notably, the LRC technique excels in situations involving degenerate pivots by reducing required iterations by 25-30%. The Greatest-Increment method also requires O(n) operations per iteration but achieves convergence with 40-50% fewer pivot operations than its counterparts. Found on pages: [ 2 ] The Steepest-Edge method emerges as the most balanced approach overall; despite its higher per-iteration cost—approximately 50% greater than basic methods—it consistently reduces necessary iterations by 30-60%, particularly benefiting large-scale problems where it can deliver solutions up to 40% faster than other evaluated approaches. All evaluations were conducted within controlled conditions in the LPBench environment to ensure clarity in performance measurements. Found on pages: [ 2 ] Practical Implications The implications of these findings offer crucial insights for implementing performance-sensitive LP solvers. By detailing each method’s strengths and weaknesses through empirical data analysis alongside theoretical foundations—including convex optimization principles—the study equips practitioners with informed guidance on selecting optimal pivot rules tailored to specific problem requirements. Found on pages: [ 24 ] Study Structure and Theoretical Insights Structured into two main sections: the first delves into Simplex algorithm implementation beginning with foundational theories while examining various pivot-selection techniques through both theoretical analyses and computational results; this section highlights their respective advantages or disadvantages comprehensively. The second section discusses mathematical concepts underlying LP problems such as duality theory while bridging abstract theory with practical applications including geometric interpretations that enhance understanding of how pivot selections relate directly back towards properties inherent within convex polytopes. Found on pages: [ 42 ] Numerical Stability Considerations Additionally addressed are practical considerations regarding numerical stability management during implementations which remain critical when applying simplex methodologies robustly across diverse contexts—from logistics through finance all way down manufacturing sectors today facing ongoing challenges optimizing simplex processes balancing efficiency against complexity levels encountered regularly throughout industry landscapes worldwide. Found on pages: [ 87 ] Conclusion and Significance In conclusion this comprehensive assessment not only elucidates primal/dual simplex mechanisms alongside pivotal selection strategies examined herein but also contributes significantly toward advancing practices surrounding linear programming applicable broadly across multiple disciplines offering meaningful guidance rooted firmly within real-world operational frameworks enhancing effectiveness overall! Found on pages: [ 342 ]
In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another, at each step improving the value of the objective function. Moreover, the method terminates after a finite number of such transitions.
RESEARCH ARTICLE A Streamlined Artificial Variable Free Version of Simplex Method
2016
This paper proposes a streamlined form of simplex method which provides some great ben-efits over traditional simplex method. For instance, it does not need any kind of artificial vari-ables or artificial constraints; it could start with any feasible or infeasible basis of an LP. This method follows the same pivoting sequence as of simplex phase 1 without showing any ex-plicit description of artificial variables which also makes it space efficient. Later in this paper, a dual version of the new method has also been presented which provides a way to easily implement the phase 1 of traditional dual simplex method. For a problem having an initial basis which is both primal and dual infeasible, our methods provide full freedom to the user, that whether to start with primal artificial free version or dual artificial free version without making any reformulation to the LP structure. Last but not the least, it provides a teaching aid for the teachers who want to teach feasibility achieveme...
Universitext, 2001
We can state now an iterative procedure for the resolution of the linear programming problem (LP) in standard form with descriptive "input data" m, n, A, band c. (m, n, A, b, C ) Step 0: