Snippets of Linear Programming

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.

The solution of a two-person zero-sum game provides optimal strategies for both players. In game-theoretic terms, the solution specifies a minimax strategy for one player and a maximin strategy for the other. These strategies can be pure strategies, that is, single strategies selected from the set ofstrategies available to each player, or they may be mixed strategies that are weighted com­ posites of the single strategies. The weights in these com­ posites consist of probabilities for playing the various strategies. The probabilities in a pair of mixed strategies are selected so that the expected value of the game is optimal for each player. The solution of a two-person zero-sum game, therefore, rests on fmding a vector of probabilities x that would allow one player to maximize the game value and another vector of probabilities y enabling the other player to minimize the game value. Solutions consisting of pure strategies can be readily obtained by examination of the game matrix, and the mixed strategies in a 2 by 2 game matrix can be deter­ mined by applying straightforward computational formulas (e.g., Coombs, Dawes, & Tversky, 1970). However, to obtain mixed-strategy solutions for game matrices with dimensionality greater than two, linear programming is required (Luce & Raiffa, 1957). Linear programming is a technique for maximizing and mini­ mizing linear functions that are subject to a number of linear constraints and nonnegativity conditions. Since the value of a game can easily be expressed as a linear function, and the rows and columns of a game matrix can serve as constraints, linear programming may be utilized to determine the vectors of probabilities x and y that, respectively, maximize and minimize the value of a two-person zero-sum game. Although linear programming computer programs have been available for some time, they are typically written for general linear programming applications. Many of the programs (such as MPOS,t for example) are large canned packages capable of solving a wide variety of linear programming problems. Since the primary application of linear programming in psychol­ ogy is for solving zero-sum games, the rationale for implementing a large general-purpose optimization package for solving a small class of problems is question­ able. These general programs require manual conversion of a game problem to a linear programming problem and manual conversion of a linear programming solution back to a game solution. Since linear programming is frequently used in business applications, many programs

Background and Study Aim. Linear and nonlinear programming are methods used to control intensity and volume in sports training. Despite their widespread application, there is a lack of evidence-based studies that directly compare the effects of linear versus nonlinear programming. This study aims to assess the effect of linear and nonlinear programming on improving the power, agility, and endurance of young basketball players. Material and Methods. This study employs a two-group pretest-posttest experimental design. It included 40 male basketball players aged 16-18, with weights ranging from 60 to 77 kg and heights from 167 to 180 cm. Participants were divided into two groups based on their ordinal pairings. The instruments used in this study were the jump DF, lane agility, and multistage fitness tests. Data were analyzed using descriptive statistics, Wilcoxon tests, and Mann-Whitney U tests, with the assistance of SPSS 23. Results. The pretest-posttest findings for both the linear and nonlinear programming groups showed significant improvements in power, speed, and endurance, with Asymptotic Significance (Asymp.sig) 2-tailed values of less than 0.05. Comparative analysis of posttest results between linear and nonlinear programming indicated significant differences: power showed an Asymp.sig value of 0.009 with a difference of 3.1; agility showed an Asymp.sig value of 0.000 with a difference of 0.35; and endurance showed an Asymp.sig value of 0.002 with a difference of 2.08. Conclusions. The study demonstrates significant differences in the impacts of linear and nonlinear programming on power, agility, and endurance among young basketball players. Both programming types significantly enhance these attributes, but nonlinear programming is more effective than linear programming in improving the physical abilities of young basketball players.

Linear bilevel programming with upper level constraints depending on the lower level solution

This study evaluates linear programming (LP) and positive mathematical programming (PMP) approaches for 3,400 farm-level models implemented in the SWISSland agent-based agricultural sector model. To overcome limitations of PMP regarding the modelling of investment decisions, we further investigated whether the forecasting performance of farm-level models could be improved by applying LP to animal production activities only, where investment in new sectors plays a major role, while applying PMP to crop production activities. The database used is the Swiss Farm Accountancy Data Network. Ex-post evaluation was performed for the period from 2005 to 2012, with the 2003-2005 three-year average as a base year. We found that PMP applied to crop production activities improves the forecasting performance of farm-level models compared to LP. Combining PMP for crop production activities with LP for modelling investment decisions in new livestock sectors improves the forecasting performance compared to PMP for both crop and animal production activities, especially in the medium and long term. For short-term forecasts, PMP for all production activities and PMP combined with LP for animal production activities produce similar results.

Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.

This essay investigates the concept of linear programming in general and linear stochastic programming in particular. Linear stochastic programming is described as the model where the parameters of the linear programming admit random variability. The first three chapters present through a set-geometric approach the foundations of linear programming. Chapter one describes the evolution of the concepts which resulted in the adoption of the model. Chapter two describes the constructs in n-dimensional euclidian space which constitute the mathematical basis of linear programs, and chapter three defines the linear programming model and develops the computational basis of the simplex algorithm. The second three chapters analyze the effect of the introduction of risk into the linear programming model. The different approaches of estimating and measuring risk are studied and the difficulties arising in formulating the stochastic problem and deriving the equivalent deterministic problems are treated from the theoretical and practical point of view. Multiple examples are given throughout the essay for clarification of the salient points.

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty. Otherwise, the robust counterpart of an uncertain program is not well defined as a robust optimization problem. In this paper, we address a key fundamental question of robust optimization: How to compute the radius of robust feasibility of uncertain linear conic programs, including linear programs? We first provide computable lower and upper bounds for the radius of robust feasibility for general uncertain linear conic programs under the commonly used ball uncertainty set. We then provide important classes of linear conic programs where the bounds are calculated by finding the optimal values of related semi-definite linear programs, among them uncertain semi-definite programs, uncertain second-order cone programs and uncertain support vector machine problems. In the case of an uncertain linear program, the exact formula allows us to calculate the radius by finding the optimal value of an associated second-order cone program.

: So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)

This new volume provides the information needed to understand the simplex method, the revised simplex method, dual simplex method, and more for solving linear programming problems.Following a logical order, the book first gives a mathematical model of the linear problem programming and describes the usual assumptions under which the problem is solved. It gives a brief description of classic algorithms for solving linear programming problems as well as some theoretical results. It goes on to explain the definitions and solutions of linear programming problems, outlining the simplest geometric methods and showing how they can be implemented. Practical examples are included along the way. The book concludes with a discussion of multi-criteria decision-making methods.Advances in Optimization and Linear Programming is a highly useful guide to linear programming for professors and students in optimization and linear programming.

Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.

This paper’s primary goal is to investigate how Contextual Education can enhancing understanding of Linear Programming. This study was entirely qualitative in nature with 15 participants. The author argues that ccontextual education techniques enhance students’ comprehension of concepts in linear programming. Students generally find linear programming difficult, especially because it is abstract and necessitates sophisticated mathematical reasoning. It seeks to close the knowledge by placing learning in realistic, real-world contexts, which promotes greater comprehension and involvement. Interviews, and group discussions were applied to learn more about how students grasp linear programming ideas, in secondary schools. Thematic analysis of the data was done with an emphasis on the conceptual clarity, problem-solving abilities, and engagement levels of the students. According to the study, it not only enhances comprehension but also boosts students’ motivation and interest, underscoring the significance of interactive and student-centered approaches in complicated mathematics topics. The study concludes by highlight the potential teaching approaches, particularly for improving comprehension of abstract ideas like linear programming. In addition to suggesting additional research to evaluate Contextual Teaching and Learning’s long-term effects on students’ mathematical competency, the study ends with suggestions for incorporating it into linear programming programs to promote deeper learning.

Microsoft Excel sensitivity analysis for linear and stochastic program feed formulation

Engineering Optimization Theory and Practice