61. Sensitivity Analysis For Linear Programming Sensitivity Analysis for linear programming. Finding linear programming offers extensive capability for addressing these questions. We http://mat.gsia.cmu.edu/QUANT/NOTES/chap8/node2.html

Next: Tableau Sensitivity Analysis Up: Quantitative Methods for Previous: Contents Sensitivity Analysis for Linear Programming Finding the optimal solution to a linear programming model is important, but it is not the only information available. There is a tremendous amount of sensitivity information , or information about what happens when data values are changed. Recall that in order to formulate a problem as a linear program, we had to invoke a certainty assumption : we had to know what value the data took on, and we made decisions based on that data. Often this assumption is somewhat dubious: the data might be unknown, or guessed at, or otherwise inaccurate. How can we determine the effect on the optimal decisions if the values change? Clearly some numbers in the data are more important than others. Can we find the ``important'' numbers? Can we determine the effect of misestimation? Linear programming offers extensive capability for addressing these questions. We begin by showing how data changes show up in the optimal table. We then give two examples of how to interpret Solver's extensive output. Tableau Sensitivity Analysis Cost Changes Right Hand Side Changes New Variable ... Exercises Michael A. Trick

62. Math Forum: Game Theory / Linear / Non-Linear Internet resources for game theory and linear or nonlinear programming Web sites, software, Internet projects, and public forums for discussion. http://mathforum.org/advanced/game.html

Game Theory, Linear/Non-Linear Programming Back to Math by Subject Math by Subject K12 Topics algebra arithmetic calculus discrete math geometry pre-calculus prob/stat Advanced Topics analysis calculus diff. equations game theory discrete math geometry (coll.) geometry (adv.) linear algebra modern algebra num. analysis This list contains some of the best resources for game theory and linear and non-linear programming. For a more exhaustive list, or to find materials that fit your specific needs, see also the Forum's Internet Mathematics Library: Operations Research. CPLEX Optimization, Inc. - ILOG, Inc. A provider of large-scale mathematical programming software and services for linear, mixed integer, and quadratic programming. Products: a linear ... David Eppstein's Teaching - David Eppstein Archives of Information and Computer Science courses in computational statistics, computability, design and analysis of algorithms, computational ... Decision Tree for Optimization Software - Hans D. Mittelmann, P. Spellucci Information on optimization software and optimization research, including links to a dictionary, tutorials, testcases, interfaces, and ...

63. Borges Home A Ruby web application framework based on Smalltalk's Seaside 2 that allows linear programming of web applications. Open source http://borges.rubyforge.org/

Sitemap Borges Home Borges is a continuation-based web application framework originally ported from on Seaside 2 that allows a linear style of programming of web applications. Components of a Borges web page can call and return from each other in a natural way, allowing complex interaction between components from simple methods. Backtracking is supported seamlessly, allowing a simple approach to building web applications that does not get in the developer's way. Borges requires Ruby 1.8 . Examples are provided, and are typically installed in /usr/local/share/examples/ruby/borges/. News Borges News RSS Feed Borges-1.1.0 Released Borges-1.1.0 fixes several memory issues allowing Borges to maintain a more stable memory size. Many Unit Tests have been added, and the Renderer API is mostly documented. You can download Borges-1.1.0 from the Borges filelist Borges Quickref Kaspar Schiess has collected this nice Borges Quick Reference document. Documentation You can read the Borges Quickref , or browse the Borges RDoc You may also be interested in the Seaside 2 documentation: Sesaide 2 Renderer API Seaside 2 Design Overview Borges-Users Mailing List If you have any questions or comments about Borges, you can

64. The Math Forum - Math Library - Linear Progrmng This page contains sites relating to linear programming. Full macro language implementation; linear programming, Goal seeking, Linear and Nonlinear more . http://mathforum.org/library/topics/linear_prog/

Browse and Search the Library Home Math Topics Operations Research : Linear Progrmng Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category Game Theory, Linear/Non-Linear Programming - Math Forum Links to some of the best Internet resources for game theory and linear or non-linear programming: Web sites, software, Internet projects, publications, and public forums for discussion. more>> Linear Programming FAQ - Optimization Technology Center, Northwestern University and Argonne National Laboratory Posted monthly to the newsgroup sci.op-research. Answers to questions such as: "What is Linear Programming?" "Where is there a good code to solve LP problems?" "Oh, and we also want to solve it as an integer program." "I wrote an optimization code. Where are some test models?" "What is MPS format?" etc. more>> All Sites - 45 items found, showing 1 to 45 Academic OR/MS Courses on the Web - Armann Ingolfsson Links to Web pages of courses in Operations Research/Management Science and related fields offered at universities or colleges all over the world. Also short courses and tutorials on OR/MS-related topics. By Armann Ingolfsson for the INFORMS Forum on ...more>> AMPL: A Modeling Language for Mathematical Programming - Bell Laboratories A comprehensive and powerful algebraic modeling language for linear and nonlinear optimization problems, in discrete or continuous variables. AMPL lets you use common notation and familiar concepts to formulate optimization models and examine solutions

65. How Efficient Is OOP In PHP? - Webmaster Stop Compares and contrasts linear programming vs OOP programming in PHP and analyzes performance results. http://www.webmasterstop.com/tutorials/efficiency-PHP-OOP.shtml

Webmaster Stop - Articles, tutorials, resources, links, and more! How efficient is OOP in PHP? Efficiency of PHP OOP by Wojciech Hlibowicki In this article, we are going to investigate the efficiency of Object Oriented Programming (OOP) in PHP, and by efficiency, we mean execution time. We are going to start by running some test cases. These test cases are going to be kept simple, since the main point of this "experiment" is to truly get the gist of how much overhead there really is when using classes. Obviously, this is not a true object oriented program, but the use of classes contribute quite a bit, and is just enough to get an idea of how efficient OOP is in PHP. Here is the code we are going to use for our first test case: OOP Function Neither class test return 1; for ($i=0; $i <1000000; $i++) return 1; for ($i=0; $i <1000000; $i++) for ($i=0; $i <1000000; $i++) $cnt+=1; Here are the execution times for the code given above. We ran through each snippet of code 10 times to get a more accurate result. First test case (in seconds) OOP Function Neither Avg * These tests were run on a Dual Xeon 2.0Ghz (using hyper-threading) with 2GB RAM.

66. Test-Problem Collection For Stochastic Linear Programming TestProblem Collection for Stochastic linear programming. http://www.uwsp.edu/math/afelt/slptestset.html

Test-Problem Collection for Stochastic Linear Programming Andy Felt , ed. Department of Mathematics and Computing University of Wisconsin-Stevens Point Brief Description This is a modern test-problem collection for stochastic programming, with emphasis on a close connection between the test problems and their associated real world applications. The problem descriptions were collected from the literature, with focus on variety of problem structure and application. Each of the 11 problems has a short description, mathematical problem statement, and notational reconciliation to a standard problem format. In addition, there are 21 specific test cases with data in SMPS format. The test set is expanding. Indeed, submissions of new problems and descriptions from colleagues are encouraged. Visit the Download Page You are welcome to freely download any and all of the test cases. Please Contribute I would love to expand the test set. If you would like to submit a problem, please email me the following: Data files in SMPS format A written description in LaTeX, following the format of the other written descriptions. (See the

67. Calipso For Linear Programming linear programming from Eric Weisstein s Encyclopedia of linear programming. see also linear programming. Bellman, Richard and Kalaba, Robert. 260 p. $?. Dantzig, George Bernard. linear programming and Extensions. http://www.uidaho.edu/~calvert/lp.html

Calipso for Linear Programming Version 3.2 of the beta test version of Calipso for linear programming is available for downloading. This version includes improvements to the integer programming algorithms. See the instructions at the bottom of this page. Please send comments and suggestions for improvement by E-mail to Jim Calvert at calvert@uidaho.edu. CALIPSO for Linear Programming is a shareware package designed for use by students and teachers of linear and integer programming. It specifically supports the textbook "Linear Programming" written by Calvert and Voxman; however, it can be used with any book. Even though it is meant for educational use, CALIPSO has considerable capability for practical application. CALIPSO will run only under Windows 95 or NT. There is an older DOS version that runs on early versions of Windows or DOS. It can be downloaded by clicking on Old DOS Version The capabilities of Calipso for Windows 95 are best described by listing its most important commands. Linear Programming Functions TABLEAU allows you to enter an LP tableau in a full-screen editor.

68. LP Explorer 1.0 LP Explorer enables the simplex method to be applied to a linear programming (LP) problem and allows the sensitivity of the solution to changes in the problem http://www.maths.ed.ac.uk/LP-Explorer/

LP Explorer 1.0 LP Explorer enables the simplex method to be applied to a linear programming (LP) problem and allows the sensitivity of the solution to changes in the problem data to be examined. LP Explorer is of particular value for problems with 2 variables when the simplex method and solution sensitivity are interpreted graphically. It is possible to enter the LP problem data via a sequence of two forms Before clicking on the box below (which runs LP Explorer for an example problem) it is suggested that you consult the user guide . This also links to further examples of LP Explorer in use and instructions on how to apply it to an LP problem of your own. LP Explorer was designed by Julian Hall (Mathematics) and Melanie Baird (Sellic On-Line), both of the University of Edinburgh. It was written by Melanie Baird in JDK1.02 and will run on any Java-enabled browser. Sellic On-Line is funded by Sun Microsystems. A review of LP Explorer 1.0 by the Java Applet Rating Service placed it in the top 25% of java-based resources. LP Explorer 1.0 has been reviewed by MERLOT.

69. HOPDM's Home Page Gondzio, J., Multiple Centrality Corrections in a PrimalDual Method for linear programming, Computational Optimization and Applications 6 (1996) 137-156. http://www.maths.ed.ac.uk/~gondzio/software/hopdm.html

HOPDM is a package for solving large scale linear, convex quadratic and convex nonlinear programming problems. The code is an implementation of the infeasible primal-dual interior point method. It uses multiple centrality correctors; their number is chosen appropriately for a given problem in order to reduce the overall solution time. HOPDM automatically chooses the most efficient factorization method for a given problem (either normal equations or augmented system). The code compares favourably with commercial LP, QP and NLP packages. HOPDM has been written by Jacek Gondzio An extension for convex QP has been developed together with Anna Altman An extension for convex NLP has been developed together with Olivier Epelly (see NLPHOPDM A decomposition environment based on HOPDM has been developed together with Robert Sarkissian . It is called PDCGM which stands for Primal-Dual Column Generation Method. Special thanks go to Marek Makowski for help in a development of the C version of the code. HOPDM has been designed to satisfy two complementary goals: to offer its user a facility of building his own interior point based application; and

70. Teaching Linear Programming With Excel Solver (CHEER V9 N3) The Virtual Edition. Volume 9, Issue 3, 1995. Teaching linear programming using Microsoft Excel Solver. The problem is formulated as the following linear program http://www.economics.ltsn.ac.uk/cheer/ch9_3/ch9_3p07.htm

The Virtual Edition Volume 9, Issue 3, 1995 Teaching Linear Programming using Microsoft Excel Solver Ziggy MacDonald University of Leicester Linear programming (LP) is one of the most widely applied O.R. techniques and owes its popularity principally to George Danzig's simplex method (Danzig 1963) and the revolution in computing. It is a very powerful technique for solving allocation problems and has become a standard tool for many businesses and organisations. Although Danzig's simplex method allows solutions to be generated by hand, the iterative nature of producing solutions is so tedious that had the computer never been invented then linear programming would have remained an interesting academic idea, relegated to the mathematics classroom. Fortunately, computers were invented and as they have become so powerful for so little cost, linear programming has become possibly one of the most widespread uses for a personal PC. There are of course numerous software packages which are dedicated to solving linear programs (and other types of mathematical program), of which possibly LINDO, GAMS and XPRESS-MP are the most popular. All these packages tend to be DOS based and are intended for a specialist market which requires tools dedicated to solving LPs. In recent years, however, several standard business packages, such as spreadsheets, have started to include an LP solving option, and Microsoft Excel is no exception. The inclusion of an LP solving capability into applications such as Excel is attractive for at least two reasons. Firstly, Excel is perhaps the most popular spreadsheet used both in business and in universities and as such is very accessible. Second to this, the spreadsheet offers very convenient data entry and editing features which allows the student to gain a greater understanding of how to construct linear programs.

71. CS 525 - Linear Programming Methods CS 525 linear programming Methods. General Course Information. This course is offered each Fall and Spring semester. CS 525 Pages of the Various Instructors. http://www.cs.wisc.edu/areas/math-prog/cs525-all.html

CS 525 - Linear Programming Methods General Course Information This course is offered each Fall and Spring semester. CS 525 Pages of the Various Instructors Michael C. Ferris (Fall '97) O.L. Mangasarian (Fall '98) Graduate MP Courses at Wisconsin CS 635 Tools and Environments for Optimization CS 719 Network Flows CS 720 Integer Programming CS 726 Nonlinear Programming Theory ... CS 733 Computational Large Sparse Systems Last modified: January 25, 1996

72. Solving Linear Programming Problems Solving linear programming Problems. http://home.xnet.com/~fidler/triton/math/review/mat124/lpsolve/lpsolv1.htm

Solving Linear Programming Problems Linear Programming Models Method Graphical Solution: 2 variables Simplex Method Return to Finite Math Topics

73. Abstract Las Vegas algorithms for linear and integer programming when the dimension is small. This paper gives an algorithm for solving linear programming problems. http://cm.bell-labs.com/who/clarkson/lp2.html

K. L.Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small Journal of the ACM , 42(2):488499, 1995. Preliminary version in Proc. 29th Annual IEEE Symposium on Foundations of Computer Science full paper source code Abstract: This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected log n)O(d) d/2+O(1) +O(d sqrt (n) log n) arithmetic operations, as n goes to infinity. The constant factors do not depend on d . Also, an algorithm is given for integer linear programming. Let phi bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let n and d be as before. Then the algorithm requires expected O(2 d dn+8 d ln ln n) +d O(d) phi ln n operations on numbers with d O(1) phi bits, as n goes to infinity, where the constant factors do not depend on d or phi . The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any given input. The technique can be extended to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of n points in E d has the same time bound.

74. Linear Programming linear programming. Michael L. Overton. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. http://www.cs.nyu.edu/cs/faculty/overton/g22_lp/encyc/article_web.html

Next: About this document Linear Programming Michael L. Overton Draft for Encyclopedia Americana December 20, 1997 LINEAR PROGRAMMING , a specific class of mathematical problems, in which a linear function is maximized (or minimized) subject to given linear constraints. This problem class is broad enough to encompass many interesting and important applications, yet specific enough to be tractable even if the number of variables is large. History. Overview. The general form of a linear program is Here and are given numbers, and are variables whose values are to be determined, maximizing the given objective subject to the given constraints. There are n variables and m constraints, in addition to the nonnegativity restrictions on the variables. The constraints are called linear because they involve only linear functions of the variables. Quadratic terms such as or are not permitted. If minimization is desired instead of maximization, this can be accomplished by reversing the signs of An example is very helpful. Consider the linear program

75. Linear Programming - An Application Of Systems linear programming. This site school texts. A sample linear programming Problem.  A linear programming Application Radio Shack Beaters. http://www.columbia.edu/~umk1/linprog.html

Linear Programming This site includes: A sample linear programming problem Suggested prerequisites For more information write to: umk1@columbia.edu Linear Programming Linear Programming, like many applications of math, was developed and used in defense in the beginning in the 1940s. It is now used in many fields especially in areas of business. I first came across linear programming in a 1972 edition of Dolciani's Algebra 1. The topic seems to be resurfacing in several secondary school texts. A sample Linear Programming Problem A Linear Programming Application: Radio Shack Beaters You are the assistant manager of an appliance store. The manager has asked you to do a cost analysis to figure out what stereo systems the store should order. Next month you will order two types of stereo systems, a less expensive Model A and a more expensive Model B. As assistant manager you must figure out how much of each model to order to minimize costs. You expect to sell at least 100 units some Model A and some Model B. Model A leaves a $40 profit for the store. Model B leaves a $60 dollar profit for the store. Total profits must be at least $4800. The wholesale cost of Model A is $250 dollars. The wholesale cost of model B is $400. As a store you by at the wholesale cost. What does a point in the solution region represent? How does it compare to a point not in your solution region? 

76. Linear Optimization A short history of linear programming In 1762, Lagrange solved tractable optimization problems with simple equality constraints. linear programming. http://home.ubalt.edu/ntsbarsh/opre640a/partVIII.htm

Deterministic Modeling: Linear Optimization with Applications Asia-Pacific Mirror Site Europe Mirror Site USA Site Para mis visitantes del mundo de habla hispana, este sitio se encuentra disponible en español en: http://home.ubalt.edu/ntsbarsh/opre640S/SpanishD.htm A mathematical optimization model consists of an objective function and a set of constraints in the form of a system of equations or inequalities. Optimization models are used extensively in almost all areas of decision-making, such as engineering design and financial portfolio selection. This site presents a focused and structured process for optimization problem formulation, design of optimal strategy, and quality-control tools that include validation, verification, and post-solution activities. Professor Hossein Arsham To search the site , try E F ind in page [Ctrl + f]. Enter a word or phrase in the dialogue box, e.g. " parameter " or " linear " If the first appearance of the word/phrase is not what you are looking for, try F ind Next MENU Optimization-Modeling Process Ingredients of Optimization Problems and Their Classification Linear Programming (LP) Dual Problem: Its Construction and Meanings ... Europe Mirror Site Companion Sites: Success Science Asia-Pacific Mirror Site Europe Mirror Site Versión en Español ... Europe Mirror Site Ingredients of Optimization Problems and Their Classification Introduction Bilevel Optimization Combinatorial Optimization Constraint Satisfaction ... Online Optimization Linear Programming (LP) Introduction LP Problem Formulation Process and Its Applications The Carpenter's Problem: Allocating scarce resources among competitive means ... How to Solve a Linear System of Equations by LP Solvers?

77. Linear Programming Packages ASLO, Neculai Andrei, ICI, Romania Mircea Barbulescu, Canada, linear programming, Simplex method. 2001 linear programming Software Survey by Robert Fourer http://www.ici.ro/camo/hlp.htm

This page is continuously updated! Last modified : January 21, 1999 Name of the Package Author(s) Purpose APOS Erling D. Andersen, TU Delft Knud D. Andersen, Dash Associates Linear and Quadratic Programming, Interior Point methods ASLO Neculai Andrei, ICI, Romania Mircea Barbulescu, Canada Linear Programming, Simplex method BPMPD Csaba Meszaros, Hungarian Academy of Sciences Linear Programming, Interior Point method CPLEX CPLEX a division of ILOG Linear Programming, Simplex / Interior Point methods FortMP Gautam Mitra, Mathematical Programming Group, Brunel University Large Scale Linear Programming, Integer Programming, Variable Separable Programming and Quadratic Programming. GULF Erik B. Bajalinov, University of L. Kossuth, Hungary. David J. Pannell, The University of Western Australia, Australia. Linear Programming, Linear-fractional Programming, Simplex method HOPDM Jacek Gondzio, The University of Edinburgh Anna Altman, Systems Research Institute, Poland Linear Programming, Interior Point method LIPSOL Yin Zhang, Rice University Linear Programming, Interior Point method LOQO Robert J. Vanderbei

78. 2D Linear Programming Applet linear programming and Pivoting in 2D. What to do (in this order) (1) Find the Input Constraints area either load an example http://cgm.cs.mcgill.ca/~beezer/cs601/main.htm

Linear Programming and Pivoting in 2D What to do (in this order): (1) Find the "Input Constraints" area - either load an example or input your own inequalities one by one - set the objective function as you wish (2) Find the dictionary area - select x1 under "enter", select xi (i > 2) under "leave" and press Pivot (3) Find the display area - click on the red vertex (this forces x2 to enter) (4) Pivot and Pivot and ... Pivot: - graphically by clicking on intersection points that share a line the red intersection point - or by selecting variables under "Enter" and "Leave" and press Pivot Click Here for Documentation Click Here for the Source files Bohdan's Geometry Page Updated October 2002

79. Course On Algorithms And Linear Programming, April 2002 - July 2002 Course on Algorithms and linear programming, April 2002 July 2002. by Uri Feige. Lecture notes Lecture 5, 15 May 2002 linear programming Duality. http://www.wisdom.weizmann.ac.il/~feige/lp02.html

Course on Algorithms and Linear Programming, April 2002 - July 2002 by Uri Feige Lecture notes: Lecture 1, 10 April 2002: The Linear Algebra of Linear Programs. Lecture 2, 24 April 2002: The Geometry of Linear Programs. Lectures 3 and 4, 1 May and 8 May 2002: The Simplex Algorithm. Lecture 5, 15 May 2002: Linear Programming Duality. Lecture 6 (by Moni Naor), 22 May 2002: Diameter of polytopes, and subexponential randomized versions of the simplex algorithm. No lecture notes. See the paper "The simplex algorithm and simple polytopes", by Gil Kalai, Math. Prog. (ser. B) 79 (1997) 217-234. Lectures 7 and 8, 29 May and 5 June 2002: The Ellipsoid Algorithm. Lecture 9, 12 June 2002, + June 19: Some more on LP duality. Lectures 10 and 11, 19 June and 26 June 2002: Perfect graphs and the theta function of Lovasz. Lectures 12 and 13 (by Moni Naor), 3 July and 10 July 2002: Lattice problems. Handouts (homework) for course. Handout 1, 10 April 2002: Introduction to Linear Programming. Handout 2,

80. BPMPD Home Page linear programming Test Problems. I m collecting linear programming problems (in MPS format, formulated as minimization) for testing. http://www.sztaki.hu/~meszaros/bpmpd/

BPMPD Home Page This page is devoted to the interior point solver called BPMPD. The solver is developed by at the MTA SZTAKI, Computer and Automation Research Institute, Hungarian Academy of Sciences Budapest Hungary ABOUT BPMPD Newest version: 2.21 (June, 1998) New features: New sparsity heuristics More efficient presolve Convex QP extension Warm-start BPMPD is a state-of-the-art implementation of a primal-dual interior point algorithm, written in C programming language. Recent version is 2.21. You can find a performance comparison of differnent IPM packages (including BPMPD) on the pages of Decision Tree for Optimization Software . Press here to view the BPMPD readme file There is a downloadable Windows95/NT executable and DLL version of the recent release. The standard BPMPD package has the following features: Standard MPS input Advanced presolve, which: removes empty rows/columns removes singleton rows eliminates free (or implied free) singleton columns removes redundant constraints and fixes variables based on minimum/maximum constraints activity does the above on the dual problem substitutes duplicated variables removes redundant bounds on variables eliminates free (or implied free) variables makes the constraint matrix sparser Sophisticated heuristic to make decision between normal equation and augmented system approach Advanced sparsity analyzer for sparse quadratic problems Scaling the problem for better numerical properties Advanced symbolic ordering for normal equations approach Left-looking supernodal factorization algorithm with loop unrolling

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