Solution Of Linear Programming Problem Research Articles

Robust optimality analysis of non-degenerate basic feasible solutions in linear programming problems with fuzzy objective coefficients

The necessarily optimal solution is known as the most reasonable solution to linear programming problems with interval/fuzzy objective function coefficients. As it remains optimal against the certain fluctuations of objective function coefficients, the necessarily optimal solution can be seen as a robust optimal solution. In this paper, we demonstrate that the necessary optimality degree of a non-degenerate basic feasible solution can be obtained easily by utilizing the tolerance approach. The necessary optimality degree evaluates to what extent the solution remains optimal against the fluctuations of objective function coefficients. Several types of fuzzy subsets showing the possible range of the objective function coefficient vector are considered. For each type of fuzzy subset, an efficient calculation method of necessary optimality degree is proposed. Numerical examples are given to illustrate the proposed methods.

ВИКОРИСТАННЯ ЕКОНОМІКО-МАТЕМАТИЧНИХ МЕТОДІВ В АНАЛІЗІ ОПТИМАЛЬНОГО РОЗВ’ЯЗКУ ЛІНІЙНОЇ ЗАДАЧІ ПЛАНУВАННЯ ВИРОБНИЦТВА

The linear problem of production planning at enterprises of the food industry is reviewed in the article. It is proved that in order to make a management decision it is necessary to perform a comprehensive analysis of the obtained solution under the influence of changing both internal and external factors. It is determined that using the theory of duality allows having a deeper analysis of the optimal solution obtained using the iterative simplex method. This algorithm, which is based on the reports generated in the MS Excel spreadsheet when solving linear problems by the simplex method, allows you to build an inverse matrix. The constructed matrix is used to determine the impact of changes in scarce resources and constraints on the optimal solution of the problem of production planning. The algorithm consists of the following stages: reduction of the linear model of production planning to the canonical form; determination of the order of basic variables of the last simplex table on the basis of reports based on results and stability and taking into account the rules of the simplex method of solving linear problems; calculating the inverse matrix and checking it for compliance with the help of the basic formula for finding the solution of a dual problem. The possibility of using the described algorithm with the example of the problem of planning the production of bakery products, where the solution was obtained by the iterative simplex method, is proved. The range of products consisted of six types of products, which are limited by the lower and upper limits of demand. Production is limited by raw materials in the form of total value, which was obtained for the planned production of certain products. The capacity of two furnaces were also set as restrictions. Investigation shows that the obtained inverse matrix of the proposed algorithm corresponds to the matrix obtained using the simplex method. The obtained results allows us to assess the impact of changes off the constraints on the solution of the linear problem of planning of the production of bakery products. It is discovered that this algorithm can be effectively used in practical work in order to conduct a deeper analysis of the optimal solutions of linear programming problems.

Solution of linear programming problem with trapezoidal fuzzy coefficients using score functions

Solution of linear programming problem with trapezoidal fuzzy coefficients using score functions

The method of dynamic calculations of elements of thermal power plants, which reduces the solution of systems of differential equations in partial derivatives to the solution of linear programming problems

The method of dynamic calculations of elements of thermal power plants, which reduces the solution of systems of differential equations in partial derivatives to the solution of linear programming problems

Clarification of the Formalization Process for Models of Economic Dynamics in the Study of the Development of Organizational and Technological Systems

The article substantiates the creation and practical development of organizational and technological systems in high-tech production on the basis of the main theory and solution of linear programming problems. The main problem when using models of economic dynamics in creating this type of system is the large dimension of the emerging optimization problems, which prevents the direct use of known computational procedures for solving linear programming problems. To do this, we consider an extreme problem for direct dynamic analysis of the Leontief model using the standard linear programming problem. At the same time, we should not forget about the problem of determining the target functional, the choice of which prevents the choice of the optimal trajectory of development of the simulated system. Due to these difficulties, the need for quality research methods of optimal trajectories to obtain theoretical conclusions for making scientifically sound planning decisions is determined. Theorems on highways describe the qualities of invariance of optimal trajectories of extreme dynamic problems in relation to the purpose of the target functional and to the choice of the optimal trajectory. The theoretical generalizations of Samuelson’s main theory were used in a consulting project on planned studies of the introduction of organizational and technological system for the production of high-tech products at a machine-building enterprise. Keywords system, technology, technology, highway, model, trajectory, programming, functional, vector, function, matrix.

VaLiPro: Linear Programming Validator for Cluster Computing Systems

The article presents and evaluates a scalable algorithm for validating solutions of linear programming problems on cluster computing systems. The main idea of the method is to generate a regular set of points (validation set) on a small-radius hypersphere centered at the point of the solution under validation. The objective function is calculated for each point of the validation set that belongs to the feasible region. If all these values are less than or equal to the value of the objective function at the point under validation, then this point is the correct solution. The parallel implementation of the VaLiPro algorithm is performed in C++ through the parallel BSF-skeleton, which encapsulates all aspects related to the MPI-based parallelization of the program. We provide the results of large-scale computational experiments on a cluster computing system to study the scalability of the VaLiPro algorithm.

Implementing blopmatching in Stata

The blopmatching estimator for average treatment effects in observational studies is a nonparametric matching estimator proposed by Díaz, Rau, and Rivera (2015, Review of Economics and Statistics 97: 803–812). This approach uses the solutions of linear programming problems to build the weighting schemes that are used to impute the missing potential outcomes. In this article, we describe blopmatch, a new command that implements these estimators.

Measures of global sensitivity in linear programming: applications in banking sector

The paper examines the sensitivity for the solution of linear programming problems using Bayesian techniques, when samples for the coefficients of the objective function are uncertain. When data is available, we estimate the solution of the linear program and provide statistical measures of uncertainty through the posterior distributions of the solution in the light of the data. When data is not available, these techniques examine the sensitivity of the solution to random variation in the coefficients of the linear problem. The new techniques are based on two posteriors emerging from the inequalities of Karush–Kuhn–Tucker conditions. The first posterior is asymptotic and does not require data. The second posterior is finite-sample-based and is used whenever data is available or if random samples can be drawn from the joint distribution of coefficients. A by-product of our framework is a robust solution. We illustrate the new techniques in two empirical applications to the case of uncertain Data Envelopment Analysis efficiency, involving two large samples, of US commercial banks and a sample of European commercial banks regulated by the Single Supervisory Mechanism. We analyse whether some pre-determined criteria, associated with size and new supervisory framework, can adequately affect the solution of linear program. The results provide evidence of substantial improvements in statistical structure with respect to sensitivities and robustification. Our methodology can serve as a consistency check of the statistical inference for the solution of linear programming problems in efficiency under uncertainty in data.

Using machine learning to predict the number of alternative solutions to a minimum cardinality set covering problem

Although the characterization of alternative optimal solutions for linear programming problems is well known, such characterizations for combinatorial optimization problems are essentially non-existent. This is the first article to qualitatively predict the number of alternative optima for a classic NP-hard combinatorial optimization problem, namely, the minimum cardinality (also called unicost) set covering problem (MCSCP). For the MCSCP, a set must be covered by a minimum number of subsets selected from a specified collection of subsets of the given set. The MCSCP has numerous industrial applications that require that a secondary objective is optimized once the size of a minimum cover has been determined. To optimize the secondary objective, the number of MCSCP solutions is optimized. In this article, for the first time, a machine learning methodology is presented to generate categorical regression trees to predict, qualitatively (extra-small, small, medium, large, or extra-large), the number of solutions to an MCSCP. Within the machine learning toolbox of MATLAB®, 600,000 unique random MCSCPs were generated and used to construct regression trees. The prediction quality of these regression trees was tested on 5000 different MCSCPs. For the 5-output model, the average accuracy of being at most one off from the predicted category was 94.2%.

Solution of Linear Programming Problem with Trapezoidal Fuzzy Coefficients using Score functions

Solution of Linear Programming Problem with Trapezoidal Fuzzy Coefficients using Score functions

Flux-corrected transport for scalar hyperbolic conservation laws and convection-diffusion equations by using linear programming

Flux-corrected transport (FCT) is one of the flux limiter methods. Unlike the total variation diminishing methods, obtaining the known FCT formulas for computing flux limiters is not quite transparent, and their transformation is not obvious when the original differential operator changes. We propose a novel formal mathematical approach to design flux correction for weighted hybrid difference schemes by using linear programming. The hybrid scheme is a linear combination of a monotone scheme and a high order scheme. The determination of maximal antidiffusive fluxes is treated as an optimization problem with a linear objective function. To obtain constraints for the optimization problem, inequalities that are valid for the monotone difference scheme are applied to the hybrid difference scheme. The numerical solution of the nonlinear optimization problem is reduced to the iterative solution of linear programming problems. A nontrivial approximate solution of the corresponding linear programming problem can be treated as the required flux limiters. We present flux correction formulas for scalar hyperbolic conservation laws and convection-diffusion equations. The designed flux-corrected transport for scalar hyperbolic conservation laws yields entropy solutions. Numerical results are presented.

Modeling of Gauss Elimination Technique and AHA Simplex Algorithm for Multi-objective Linear Programming Problems

In this research paper, an effort has been made to solve each linear objective function involved in the Multi-objective Linear Programming Problem (MOLPP) under consideration by AHA simplex algorithm and then the MOLPP is converted into a single LPP by using various techniques and then the solution of LPP thus formed is recovered by Gauss elimination technique. MOLPP is concerned with the linear programming problems of maximizing or minimizing, the linear objective function having more than one objective along with subject to a set of constraints having linear inequalities in nature. Modeling of Gauss elimination technique of inequalities is derived for numerical solution of linear programming problem by using concept of bounds. The method is quite useful because the calculations involved are simple as compared to other existing methods and takes least time. The same has been illustrated by a numerical example for each technique discussed here.

Optimizing the Material Fluxes in Flexible Production Systems with Interchangeable Machining Equipment

The material fluxes in the machining sections of flexible production systems may be optimized by appropriate organization of the equipment, on the basis of simulated annealing and a simplex method for the solution of linear-programming problems.

On the solution of linear programming problems with rough interval coefficients in a fuzzy environment

On the solution of linear programming problems with rough interval coefficients in a fuzzy environment

Complete Solution for Linear Programming Problem in two Variables and One Restriction

Abstract In this work we consider an elementary linear programming problem with one restriction in order to be returned the set of solutions. For this, we include the corresponding source code.

A Novel Approach for Solving Quadratic Fractional Programming Problems

In this paper, the quadratic fractional programming (QFP) problems involving a factorized or not-factorized objective function and subject to homogenous or non-homogenous constraints is considered. Our proposed approach depends on a computational method that converts QFP problem into a linear programming (LP) problem by using a Taylor series to solve the problem algebraically. This approach, based on the solution of LP problems can be applied to various types of of nonlinear fractional programming problems containing nonlinear constraint(s) and minimizes the total execution time on iterative operations. To illustrate the solution process, two examples are presented and the proposed approach is compared with other two existing methods for solving QFP problems.

Two simple ways to find an efficient solution for a multiple objective linear programming problem

Abstract A number of methods and techniques for determining “effective” solutions for multiple objective linear programming problems (MPP) have been developed. In this study, we will present two simple methods for determining an efficient solution for a MPP that reducing the given problem to a one-objective linear programming problem. One of these methods falls under the category of methods of weighted metrics, and the other is an approach similar to the ε- constraint method. The solutions determined by the two methods are not only effective and are found on the Pareto frontier, but are also “the best” in terms of distance to the optimal solutions for all objective function from the MPP. Obviously, besides the optimal solutions of linear programming problems in which we take each objective function, we can also consider the ideal point and Nadir point, in order to take into account all the notions that have been introduced to provide a solution to this problem

A hybrid isotonic separation training algorithm with correlation-based isotonic feature selection for binary classification

Isotonic separation is a classification technique which constructs a model by transforming the training set into a linear programming problem (LPP). It is computationally expensive to solve large-scale LPPs using traditional methods when data set grows. This paper proposes a hybrid binary classification algorithm, meta-heuristic isotonic separation with particle swarm optimization and convergence criterion (MeHeIS–CPSO), in which a particle swarm optimization-based meta-heuristic is embedded in the training phase to find a solution for LPP. The proposed framework formulates the LPP as a directed acyclic graph (DAG) and arranges decision variables using topological sort. It obtains a new threshold value from training set and sets up a convergence criterion using this threshold. It also deploys a new correlation coefficient-based supervised feature selection technique to select isotonic features and improves predictive accuracy of the classifier. Experiments are conducted on publicly available data sets and synthetic data set. Theoretical, empirical, and statistical analyses show that MeHeIS–CPSO is superior to its predecessors in terms of training time and predictive ability on large data sets. It also outperforms state-of-the-art machine learning and isotonic classification techniques in terms of predictive performance on small- and large-scale data sets.

Statistical Averaging Method and New Statistical Averaging Method for Solving Extreme Point Multi–Objective Linear Programming Problem

In this paper, statistical averaging method (arithmetic mean, geometric mean) and new statistical averaging method (new arithmetic mean, new geometric mean) have been proposed for extreme point multi-objective linear programming problem (EPMOLPP). Extreme point can be taken from graphical representation of linear programming problem (LPP). Graphical solution of LPP has been discussed in this research The objective of this method is for making single objective from multi-objective extreme point linear programming problem. Chandra Sen’s method is for making single objective from multi-objective linear programming problem (MOLPP). Here Chandra Sen’s method has also been used to solve EPMOLPP. An algorithm and program solution have been given for our proposed method to solve such type of problems. A numerical example is given and the result in Table 2 indicates that the proposed technique gives better results.

A Pivot Rule for Maximization Degeneracy Problems of Simplex Method for Linear Programming Problems

The simplex algorithm is a powerful technique, widely used to find the optimal solution of linear programming problems. Pivot rule is the most important first step of the simplex algorithm in the maximization degeneracy problems of LP for selecting the entering variable/work column (to obtain the smallest ratios) and after than leaving variable for pivot equation. The selection of an effective pivot rule leads the smallest number of iterations of the optimal solution of L.P. The selecting of most negative number (for entering variable) in the maximization problems is known as G.B Dantzig's pivot rule. The purpose of this paper is to model an algorithm that improves in the entering and leaving variable for the maximization degeneracy problems of LP. In this article, we have introduced a powerful technique for pivot rule which reduces the number of iterations to obtain the optimal solution as compare to Dantzig’s pivot rule. This article gives better concept of selection the entering variable as well leaving variable. This article also gives a helpful imminent into the unique and constructive performance of the proposed method by coverage computational experiments.