A procedure for choosing incoming and exiting variables in the simplex algorithm

Linear programming algorithms have been widely used in Decision Support Systems. These systems have incorporated linear programming algorithms for the solution of the given problems. Yet, the special structure of each linear problem may take advantage of different linear programming algorithms or different techniques used in these algorithms. This paper proposes a web-based DSS that assists decision makers in the solution of linear programming problems with a variety of linear programming algorithms and techniques. Two linear programming algorithms have been included in the DSS: (i) revised simplex algorithm and (ii) exterior primal simplex algorithm. Furthermore, ten scaling techniques, five basis update methods and eight pivoting rules have been incorporated in the DSS. All linear programming algorithms and methods have been implemented using MATLAB and converted to Java classes using MATLAB Builder JA, while the web interface of the DSS has been designed using Java Server Pages.

The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most violated reduced cost. In their seminal work, Klee and Minty showed that this pivot rule takes exponential time in the worst case. We prove two main results on the simplex method. Firstly, we show that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzig's pivot rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a specific variable to enter the basis is PSPACE-complete. We use the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs. We construct MDPs and then show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method.

*† This paper proposes a new algorithm for actuator allocation problems, e.g., reaction wheel torque distribution and jet selection logic. The algorithm is based on an interior-point linear programming (LP) algorithm. Compared to the simplex LP algorithm, the interiorpoint LP algorithm exhibits several advantages at the cost of reasonably increased computation. They encompasses greater robustness to uncertainty in actuator models, smoother actuation command over time, and greater flexibility in incorporating nonlinear operations such as quantization. These advantages stem from the fact that the interior algorithm searches for a suboptimal solution among the interior points of a feasible set and not among the extreme points or corner points of the feasible set as in the simplex algorithm. The proposed approach has been applied to the problem of reaction wheel torque distribution for the attitude control of an autonomous rendezvous and docking (ARD) robotic spacecraft and demonstrated to yield more effective solutions for the actuator allocation problem than a simplex LP approach.

In this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

The main viewpoint of land use planning is to resolve the conflict between competing land uses in one place that is carried out based on their economic, social and environmental suitability. Many different approaches have been designed to resolve this conflict but any of them have done. This study aimed to assess accuracy and validity compared four optimization algorithms, including linear programming (LP), simulated annealing (SA), multi-objective land use allocation (MOLA) and multidimensional choice (MDCHOICE) for land use planning in Gorgan Township. At first, land use allocation was done through the LP algorithm. Then, using achieved pixels quantity from LP optimization, the land uses was optimized in term of spatial suitability by other algorithms. The allocation maps generated through these algorithms were compared using statistical methods and landscape metrics. The map provided through MDCHOICE was used as a reference map and that generated through MOLA, SA and LP as a comparative one in a validation process. The similarities and agreements of the two maps were evaluated with various algorithms in IDRISI and MCK software. The results demonstrate that: (1) It follows that the LP algorithm to determine the optimal number of pixels to lead MDCHOICE has helped in the spatial allocation properly. (2) MOLA had a better performance for development in terms of all landscape metrics and SA algorithm was better in forestry land use allocation. Therefore the quality of spatial allocation both of them needs to be improved. (3) Although the LP algorithm is not superior to the algorithms based on the dimensions studied in this research, in terms of achieving the defined goals and the degree of deviation from the number of cells defined for each land use has the least violation. (4) The superiority of MDCHOICE over MOLA, SA and LP and applicability of various kappa statistics for comparing the maps for a more in depth analysis of agreements and disagreements.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

We present three new pivot rules for the Simplex method for Linear Programs over 0/1-polytopes. We show that the number of nondegenerate steps taken using these three rules is strongly polynomial, linear in the number of variables, and linear in the dimension. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1-polytopes and novel modifications to the classical steepest-edge and shadow-vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization. Funding: A. E. Black and J. A. De Loera are grateful for the support received through the National Science Foundation [Grants DMS-1818969 and NSF GRFP]. L. Sanita is grateful for the support received from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant VI.Vidi.193.087].

<p>Optimization has become an important tool in treatment planning for cancer radiation therapy. It may be used to determine beam weights, beam directions, and appropriate use of beam modifiers such as wedges and blocks, with the aim of delivering a required dose to the tumor while sparing nearby critical structures and normal tissue. Linear programming formulations are a core computation in many approaches to treatment planning, because of the abundance of highly developed linear programming software. Moreover the choices of formulation, algorithm, and pivot rule that perform best from a computational view point are sometimes not obvious, and the software’s default choices are sometimes poor. Here we present some linear programming formulations of treatment planning problem with dose volume constraints, conclusions are drawn about the formulations and variants.</p>

In this paper, we show that Ye‐Todd‐Mizuno's O(\(\sqrt {nL} \)) ‐iteration homogeneous andself‐dual linear programming (LP) algorithm possesses quadratic convergence of the duality gap to zero. In the case of infeasibility, this shows that a homogenizing variable quadratically converges to zero (which proves that at least one of the primal and dual LP problems is infeasible) and implies that the iterates of the (original) LP variable quadratically diverge. Thus, we have established a complete asymptotic convergence result for interior‐point algorithms without any assumption on the LP problem.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

This paper proposes a novel energy management approach (imitation-Q-learning) based on imitation learning (IL) and reinforcement learning (RL). The proposed approach reinforces a decision-making agent based on a modified Q-learning algorithm to mimic an expert demonstration to solve a microgrid (MG) energy management problem. Those demonstrations are derived from solving a set of linear programming (LP) problems. Consequently, the imitation-Q-learning algorithm learns by interacting with the MG simulator and imitating the LP demonstrations to make decisions in real time that minimize the MG energy costs without prior knowledge of uncertainties related to photovoltaic (PV) production, load consumption, and electricity prices. A real-scale MG at the Lille Catholic University in France was used as a case study to conduct experiments. The proposed approach was compared to the expert performances, which are the LP algorithm and the conventional Q-learning algorithm in different test scenarios. It was approximately 80 times faster than conventional Q-learning and achieved the same performance as LP. In order to test the robustness of the proposed approach, a PV inverter crush and load shedding were also simulated. Preliminary results show the effectiveness of the proposed method.

Simplex Method is one of the most popular methods to solve Linear Programming problems. Many rules and modifications on Simplex Method are proposed to improve the efficiency of iteration. In this study, traditional rules and latest modifications are presented and introduced, focusing either on revision on pivot rules or on method combination. These rules and modifications can surely either reduce the number of pivot steps or simplify calculation. Some of the traditional rules are further modified and even combined so that latest modifications enjoy higher efficiency. For certain methods, further rules are needed to determine the extent to which the method can improve the efficiency.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. Kheirfam and Hasani (Sensitivity analysis for fuzzy linear programming problems with fuzzy variables, Advanced Modeling and Optimization, 12 (2010) 257-272), proposed a method for the sensitivity analysis for fuzzy linear programming (FLP) problems with fuzzy variables. There are some important cases that are not considered by Kheirfam and Hasani. In this paper, those cases are considered and numerical examples are solved.