Advances in Optimization and Linear Programming

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.

A common type of mathematical optimization is Linear Programming (LP). An LP solution of aquifer influence functions has recently been reported by Gadjica, etal.1 (1987) and Targac, etal.2 Their LP matrices were large and sparse (only 3% of the elements were non-zero) and were solved on main frame computers. Another recent application of LP is equation-of-state matching of laboratory PVT data3. This problem leads to a smaller, denser LP matrix. Three methods of LP solution were investigated on microcomputers: (1) the simplex method, (2) the revised simplex method, and (3) the symmetric method. These methods were run on several LP problems ranging from a small dense matrix to large sparse matrices. The different methods have different characteristics which affect the speed, storage requirements and simplicity of coding. The simplex method is straightforward, but usually is slower and requires more storage than the other methods. The results of this study are tabulated with running times and storage requirements for the various LP methods and microcomputers. The computers range from the IBM XT to the Compaq 386. This information serves as a documentation of the LP codes and should be useful for an engineer interested in using LP codes on a microcomputer.

Elementary Linear Programming with Applications

In Operations Research, linear-fractional programming is considered as the generalization of linear programming problem. While in a linear programming the objective function is a linear function, and in a linear-fractional programming the objective function is the ratio of two linear or non-linear functions. The majority of the algorithm used for solving the linear fractional programming problem relies upon the classical simplex method. In this paper, we have proposed a new algorithm for solving a linear fractional programming problem in which the objective function is a combination of linear fractional function, while constraint functions are in the form of linear inequalities. Our proposed algorithm is based on the extension of the method, which is used to solve linear programming problems with linear constraints. The primary intent behind developing this method is that we did not need to transform the linear fractional programming problem into linear programming problem, and also it helps in finding out the feasible region via a sequence of points in the direction that improves the feasibility of the fractional objective function. Numerical examples are given to illustrate the use of these proposed methods. Lastly, to demonstrate the efficacy of the proposed algorithm, we have compared the findings obtained with other approaches to display our algorithm's efficacy.

Chapter two - Linear programming

An efficient algorithm is proposed for finding all solutions of piecewise‐linear (PWL) resistive circuits using linear programming (LP). This algorithm is based on a simple test (termed the LP test) for non‐existence of a solution to a system of PWL equations in a given region. In the conventional LP test, the system of PWL equations is transformed into an LP problem, to which the simplex method is applied. However, this algorithm requires a very large number of pivotings because the simplex method is applied on many regions. In this paper, we introduce the dual simplex method to the LP test, which makes the average number of pivotings per region much smaller (less than one, for example) and makes the algorithm very efficient. By numerical examples, it is shown that the proposed algorithm could find all solutions of large‐scale problems, including those where the number of variables is 300 and the number of linear regions is 10300, in practical computation time. Copyright © 2002 John Wiley & Sons, Ltd.

Engineering Optimization Theory and Practice

In introducing three survey articles by Anstreicher, Monma and Wright, 1 the editor of the March 1989 SIAM News made the following comments about linear programming (LP) problems: According to current estimates, more than $100 million in human and computer time is invested yearly in the formulation and solution of linear programming problems. Businesses, large and small, use linear programming models to optimize communications systems and to schedule transportation networks, to control inventories, to plan portfolios, to maximize output... Invented in the mid-1940s by George Dantzig and improved in various ways in the intervening four decades, the simplex method continues to be the workhorse algorithm for solving linear programming problems. It is no wonder, then, that the announcement in 1984 of a method [AT&T researcher N. Karmarkar’s Projective algorithm2] with the potential for dramatic improvement in computational effectiveness over the simplex method made front-page news in major newspapers and magazines throughout the country.

Chapter 21 - The decomposition principle for linear programs

An efficient algorithm is proposed for finding all solutions of piecewise-linear (PWL) resistive circuits using linear programming (LP). This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of PWL equations in a given region. In the LP test, the system of PWL equations is transformed in to an LP problem, to which the simplex method is applied. Such an LP problem is obtained by surrounding the PWL functions by rectangles. In this paper, we introduce the dual simplex method to the LP test, which makes the average number of pivotings per region much smaller (less than one, for example) and makes the algorithm very efficient.

Hybrid-LP: Finding advanced starting points for simplex, and pivoting LP methods

This paper deals with a class of mathematical programming problems that includes linear and nonlinear programming problems in a particular form. First, a linear programming problem is considered, and the possibility of deriving its direct complete solution in terms of traditional mathematics without using known iterative computational procedures and algorithms of linear programming, such as the simplex method, is studied. Direct solutions to the problem in the case of minimal dimension with a reduced set of constraints are proposed. It is shown that the derivation of such solutions, as dimension increases, becomes a very complicated problem with increasing dimension and, therefore, is hardly feasible. Some examples of other linear and nonlinear programming problems, which can be obtained from the above-considered problem by means of isomorphic transformations, are presented. The main definitions and preliminary results of tropical mathematics, which are required for the subsequent description and application of tropical optimization methods, are then outlined. A tropical optimization problem is formulated, and direct complete solutions of this problem and of its special cases are given. The above-formulated linear and nonlinear programming problems are reduced to a tropical optimization problem to provide their direct complete solution in terms of tropical mathematics. The solution of the linear programming problem with a reduced set of constraints is written in terms of traditional mathematics.

Introduction. When formulating the classical two-stage transportation problem, it is assumed that the product is transported from suppliers to consumers through intermediate points. Intermediary firms and various kinds of storage facilities (warehouses) can act as intermediate points. The article discusses two mathematical models for two-stage transportation problem (linear programming problem and quadratic programming problem) and a fairly universal way to solve them using modern software. It uses the description of the problem in the modeling language AMPL (A Mathematical Programming Language) and depends on which of the known programs is chosen to solve the problem of linear or quadratic programming. The purpose of the article is to propose the use of AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems, to formulate a mathematical model of a quadratic programming two-stage transportation problem and to investigate its properties. Results. The properties of two variants of a two-stage transportation problem are described: a linear programming problem and a quadratic programming problem. An AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems is given. The results of the calculation using Gurobi program for a linear programming two-stage transportation problem, which has many solutions, are presented and analyzed. A quadratic programming two-stage transportation problem was formulated and conditions were found under which it has unique solution. Conclusions. The developed AMPL-code for a linear programming two-stage transportation problem and its modification for a quadratic programming two-stage transportation problem can be used to solve various logistics transportation problems using modern software for solving mathematical programming problems. The developed AMPL code can be easily adapted to take into account the lower and upper bounds for the quantity of products transported from suppliers to intermediate points and from intermediate points to consumers. Keywords: transportation problem, linear programming problem, AMPL modeling language, Gurobi program, quadratic programming problem.

Linear programming(LP) is the term used for defining a wide range of optimization problems in which the objective function to be minimized or maximized is linear in the unknown variables and the constraints are a combination of linear equalities and inequalities. LP problems occur in many real-life economic situations where profits are to be maximized or costs minimized with constraint limits on resources. While the simplex method introduced in a later reference can be used for hand solution of LP problems, computer use becomes necessary even for a small number of variables. Problems involving diet decisions, transportation, production and manufacturing, product mix, engineering limit analysis in design, airline scheduling, and so on are solved using computers. This technique is called sequential linear programming (SLP). This paper describes LP's problems and solves a LP's problems using the dual simplex method neural networks

One of the most significant and well-studied optimization problems is the Linear Programming problem (LP). Many algorithms have been proposed for the solution of Linear Programming problems (LPs); the main categories of them are: (i) simplex-type or pivoting algorithms, (ii) interior-point methods (IPM) and (iii) exterior point simplex type algorithms (EPSA). Prior to the application of these algorithms, some preconditioning techniques are executed in order to improve the computational properties of LPs. Scaling is the most widely used preconditioning technique and is used to reduce the condition number of the constraint matrix, reduce the number of iterations required to solve LPs and improve the numerical behavior of the algorithms.