Operations Research : New Applications
http://www.maplesoft.com/applications/category.aspx?cid=159
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 02 Sep 2014 21:43:33 GMTTue, 02 Sep 2014 21:43:33 GMTNew applications in the Operations Research categoryhttp://www.mapleprimes.com/images/mapleapps.gifOperations Research : New Applications
http://www.maplesoft.com/applications/category.aspx?cid=159
Circle Packing in a Square
http://www.maplesoft.com/applications/view.aspx?SID=153599&ref=Feed
<p>This application optimizes the packing of circles (of varying radii) in a square, such that the side-length of the square is minimized. One solution for 20 circles (with integer radii of 1 to 20) is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 230 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p><img src="/view.aspx?si=153599/071f7b81258c5cad651a5030370d824f.gif" alt="Circle Packing in a Square" align="left"/><p>This application optimizes the packing of circles (of varying radii) in a square, such that the side-length of the square is minimized. One solution for 20 circles (with integer radii of 1 to 20) is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 230 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>153599Wed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanPacking Circles into a Triangle
http://www.maplesoft.com/applications/view.aspx?SID=153596&ref=Feed
<p>This application finds the best packing and largest radius of equal-sized circles, such that they fit in a pre-defined triangle. One solution, as visualized by this application, is given below.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="http://www.maplesoft.com/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 310 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations. The vertices of the triangle can also be modified</p>
<p>Applications like this are used to stress-test global optimizers.</p><img src="/view.aspx?si=153596/2ac6ca1378717b3d939f3d8107616b35.gif" alt="Packing Circles into a Triangle" align="left"/><p>This application finds the best packing and largest radius of equal-sized circles, such that they fit in a pre-defined triangle. One solution, as visualized by this application, is given below.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="http://www.maplesoft.com/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 310 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations. The vertices of the triangle can also be modified</p>
<p>Applications like this are used to stress-test global optimizers.</p>153596Wed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanPacking Disks into a Circle
http://www.maplesoft.com/applications/view.aspx?SID=153600&ref=Feed
<p>This application finds the best packing of unequal non-overlapping disks in a circular container, such that the radius of the container is minimized. This is a tough global optimization problem that demands strong solvers; this application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>. You must have the Global Optimization Toolbox installed to use this application.</p>
<p>One solution for the packing of 50 disks with the integer radii 1 to 50 (as found by this application) is visualized here.</p>
<p>Other solutions for similar packing problems are documented at <a href="http://www.packomania.com">http://www.packomania.com</a>.</p>
<p>Packing optimization is industrially important, with applications in pallet loading, the arrangement of fiber optic cables in a tube, or the placing of components on a circuit board.</p><img src="/view.aspx?si=153600/32183b61c1bca332d0c71924ae09f73a.gif" alt="Packing Disks into a Circle" align="left"/><p>This application finds the best packing of unequal non-overlapping disks in a circular container, such that the radius of the container is minimized. This is a tough global optimization problem that demands strong solvers; this application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>. You must have the Global Optimization Toolbox installed to use this application.</p>
<p>One solution for the packing of 50 disks with the integer radii 1 to 50 (as found by this application) is visualized here.</p>
<p>Other solutions for similar packing problems are documented at <a href="http://www.packomania.com">http://www.packomania.com</a>.</p>
<p>Packing optimization is industrially important, with applications in pallet loading, the arrangement of fiber optic cables in a tube, or the placing of components on a circuit board.</p>153600Wed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanCircle Packing in an Ellipse
http://www.maplesoft.com/applications/view.aspx?SID=153598&ref=Feed
<p>This application optimizes the packing of circles in an ellipse, such that the area of the ellipse is minimized. A typical solution is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 35 circles generates 666 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>
<p>The constraints and ellipse parameterization are taken from "Packing circles within ellipses", Birgin et al., International Transactions in Operational Research , Volume 20, Issue 3, pages 365–389, May 2013.</p><img src="/view.aspx?si=153598/5f52383daddaeb53aec548d14ebd6ce0.gif" alt="Circle Packing in an Ellipse" align="left"/><p>This application optimizes the packing of circles in an ellipse, such that the area of the ellipse is minimized. A typical solution is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 35 circles generates 666 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>
<p>The constraints and ellipse parameterization are taken from "Packing circles within ellipses", Birgin et al., International Transactions in Operational Research , Volume 20, Issue 3, pages 365–389, May 2013.</p>153598Wed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanClassroom Tips and Techniques: Best Taylor-Polynomial Approximations
http://www.maplesoft.com/applications/view.aspx?SID=136471&ref=Feed
In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.<img src="/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" align="left"/>In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.136471Tue, 14 Aug 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: An Inequality-Constrained Optimization Problem
http://www.maplesoft.com/applications/view.aspx?SID=135904&ref=Feed
<p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p><img src="/view.aspx?si=135904/thumb.jpg" alt="Classroom Tips and Techniques: An Inequality-Constrained Optimization Problem" align="left"/><p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p>135904Mon, 16 Jul 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezStreet-fighting Math
http://www.maplesoft.com/applications/view.aspx?SID=129226&ref=Feed
This interactive Maple document contains a simple street-fighting game and performs a mathematical analysis of it, involving probability and game theory. The document is suitable for presentation in an undergraduate course on operations research, probability or linear programming. No knowledge of Maple is required.<img src="/view.aspx?si=129226/fighter_sm.jpg" alt="Street-fighting Math" align="left"/>This interactive Maple document contains a simple street-fighting game and performs a mathematical analysis of it, involving probability and game theory. The document is suitable for presentation in an undergraduate course on operations research, probability or linear programming. No knowledge of Maple is required.129226Thu, 29 Dec 2011 05:00:00 ZDr. Robert IsraelDr. Robert IsraelGreat Expectations
http://www.maplesoft.com/applications/view.aspx?SID=127116&ref=Feed
<p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p><img src="/view.aspx?si=127116/expectation_thum.png" alt="Great Expectations" align="left"/><p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p>127116Thu, 27 Oct 2011 04:00:00 ZDirectSearch optimization package, version 2
http://www.maplesoft.com/applications/view.aspx?SID=101333&ref=Feed
<p> The DirectSearch package is a collection of commands to numerically compute local and global minimums (maximums) of nonlinear multivariate function with (without) constraints. The package optimization methods are universal derivative-free direct searching methods, i.e. they do not require the objective function and constraints to be differentiable and continuous.<br /> The package optimization methods have quadratic convergence for quadratic functions.<br /><br /> The package also contains commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data.<br /><br />The following is a summary of the major improvements in DirectSearch v.2.<br /><br />-- Three new derivative-free optimization methods are added.<br />-- The new global optimization command GlobalOptima is added.<br />-- The commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data are added.<br />-- Mixed integer-discrete-continuous optimization is now supported.<br />-- You can now specify inequality constraints as any Boolean expressions.<br />-- You can now set bound inequality constraints x>=a, x<=b as: x=a..b.<br />-- Assume and assumption commands are supported for inequality constraints.<br />-- You can now specify problem variables as Vector.<br />-- High dimensional optimization problem are now solved a much faster.<br />-- Search in space curve direction is added to all algorithms.<br />-- Penalty function method is added for optimization with inequality constraints<br />-- Improved optimization algorithm for equality constraints is faster and more reliable.<br />-- The feasible initial point searching is improved.<br />-- Now the package is compatible with Maple 12 and above.<br />-- Detailed description of CDOS method in .pdf format is added.<br />-- Russian version of the package is now available.<br /><br /></p><img src="/view.aspx?si=101333/maple_icon.jpg" alt="DirectSearch optimization package, version 2" align="left"/><p> The DirectSearch package is a collection of commands to numerically compute local and global minimums (maximums) of nonlinear multivariate function with (without) constraints. The package optimization methods are universal derivative-free direct searching methods, i.e. they do not require the objective function and constraints to be differentiable and continuous.<br /> The package optimization methods have quadratic convergence for quadratic functions.<br /><br /> The package also contains commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data.<br /><br />The following is a summary of the major improvements in DirectSearch v.2.<br /><br />-- Three new derivative-free optimization methods are added.<br />-- The new global optimization command GlobalOptima is added.<br />-- The commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data are added.<br />-- Mixed integer-discrete-continuous optimization is now supported.<br />-- You can now specify inequality constraints as any Boolean expressions.<br />-- You can now set bound inequality constraints x>=a, x<=b as: x=a..b.<br />-- Assume and assumption commands are supported for inequality constraints.<br />-- You can now specify problem variables as Vector.<br />-- High dimensional optimization problem are now solved a much faster.<br />-- Search in space curve direction is added to all algorithms.<br />-- Penalty function method is added for optimization with inequality constraints<br />-- Improved optimization algorithm for equality constraints is faster and more reliable.<br />-- The feasible initial point searching is improved.<br />-- Now the package is compatible with Maple 12 and above.<br />-- Detailed description of CDOS method in .pdf format is added.<br />-- Russian version of the package is now available.<br /><br /></p>101333Tue, 01 Feb 2011 05:00:00 ZDr. Sergey MoiseevDr. Sergey MoiseevA Computational Approach to Essential and Nonessential Objective Functions in Linear Multicriteria Optimization
http://www.maplesoft.com/applications/view.aspx?SID=7061&ref=Feed
<strong>Authors</strong>: Prof. Agnieszka B. Malinowska and Prof. Delfim F. M. Torres
The question of obtaining well-defined criteria for multiple criteria decision making problems is well-known. One of the approaches dealing with this question is the concept of nonessential objective function. A certain objective function is called nonessential if the set of efficient solutions is the same both with or without that objective function. In this work we put together two methods for determining nonessential objective functions. A computational implementation is done using the computer algebra system Maple.<img src="/view.aspx?si=7061/1.jpg" alt="A Computational Approach to Essential and Nonessential Objective Functions in Linear Multicriteria Optimization" align="left"/><strong>Authors</strong>: Prof. Agnieszka B. Malinowska and Prof. Delfim F. M. Torres
The question of obtaining well-defined criteria for multiple criteria decision making problems is well-known. One of the approaches dealing with this question is the concept of nonessential objective function. A certain objective function is called nonessential if the set of efficient solutions is the same both with or without that objective function. In this work we put together two methods for determining nonessential objective functions. A computational implementation is done using the computer algebra system Maple.7061Tue, 23 Dec 2008 00:00:00 ZProf. Delfim TorresProf. Delfim TorresDijkstras Shortest Path Algorithm
http://www.maplesoft.com/applications/view.aspx?SID=4969&ref=Feed
An implementation of Dijkstra's Shortest Path algorithm as a Maple package.<img src="/view.aspx?si=4969/dijkstra_v2_maple11_4.jpg" alt="Dijkstras Shortest Path Algorithm" align="left"/>An implementation of Dijkstra's Shortest Path algorithm as a Maple package.4969Tue, 29 May 2007 00:00:00 ZJay PedersenJay PedersenClassroom Tips and Techniques: The Lagrange Multiplier Method
http://www.maplesoft.com/applications/view.aspx?SID=4811&ref=Feed
Maple has a number of graphical and analytical tools for studying and implementing the method of Lagrange multipliers. In this article, we demonstrate a number of these tools, indicating how they might be used pedagogically.<img src="/view.aspx?si=4811/lagrange.gif" alt="Classroom Tips and Techniques: The Lagrange Multiplier Method" align="left"/>Maple has a number of graphical and analytical tools for studying and implementing the method of Lagrange multipliers. In this article, we demonstrate a number of these tools, indicating how they might be used pedagogically.4811Mon, 28 Aug 2006 00:00:00 ZDr. Robert LopezDr. Robert LopezAspherical Lens Surface Identification - Non-Linear Fitting with the Global Optimization Toolbox
http://www.maplesoft.com/applications/view.aspx?SID=4806&ref=Feed
<p>In this Application Demonstration, we investigate Aspherical Lenses and apply non-linear fitting to obtain an accurate representation of the given data in the form of a function, using the GlobalOptimization Toolbox for Maple.</p><img src="/view.aspx?si=4806/asphlens.jpg" alt="Aspherical Lens Surface Identification - Non-Linear Fitting with the Global Optimization Toolbox" align="left"/><p>In this Application Demonstration, we investigate Aspherical Lenses and apply non-linear fitting to obtain an accurate representation of the given data in the form of a function, using the GlobalOptimization Toolbox for Maple.</p>4806Mon, 31 Jul 2006 04:00:00 ZMaplesoftMaplesoftCircuit Design Problem
http://www.maplesoft.com/applications/view.aspx?SID=1678&ref=Feed
Based on the classical study of Ebers and Moll (1954), a bipolar transistor is modeled by an electrical circuit (see also e.g., Granvilliers and Benhamou, 2001). The corresponding model leads to a square system if highly nonlinear equations in nine (9) variables that has been studied by numerous researchers, in attempts to solve it, and then prove the correctness of the suggested solution.<img src="/view.aspx?si=1678/EMOLL.JPG" alt="Circuit Design Problem" align="left"/>Based on the classical study of Ebers and Moll (1954), a bipolar transistor is modeled by an electrical circuit (see also e.g., Granvilliers and Benhamou, 2001). The corresponding model leads to a square system if highly nonlinear equations in nine (9) variables that has been studied by numerous researchers, in attempts to solve it, and then prove the correctness of the suggested solution.1678Wed, 26 Oct 2005 00:00:00 ZDr. Janos PinterDr. Janos PinterAlkylation Process Model
http://www.maplesoft.com/applications/view.aspx?SID=1675&ref=Feed
In this example, we describe a model for the optimization of a typical process operation in the petrochemical industry. Our objective is to determine the optimal set of operating conditions for an alkylation process that combines olefin with isobutane, in the presence of a catalyst, to form alkylate.
Many chemical processes are characterized by nonlinear equilibrium (material and energy balance) constraints. In addition, the processes are typically constrained by restrictions on the operating ranges of the decision variables such as amounts and rates of the components used, temperature, pressure, and so on.
This worksheet requires that the Global Optimization Toolbox has been added to Maple.<img src="/view.aspx?si=1675/alkyl.JPG" alt="Alkylation Process Model" align="left"/>In this example, we describe a model for the optimization of a typical process operation in the petrochemical industry. Our objective is to determine the optimal set of operating conditions for an alkylation process that combines olefin with isobutane, in the presence of a catalyst, to form alkylate.
Many chemical processes are characterized by nonlinear equilibrium (material and energy balance) constraints. In addition, the processes are typically constrained by restrictions on the operating ranges of the decision variables such as amounts and rates of the components used, temperature, pressure, and so on.
This worksheet requires that the Global Optimization Toolbox has been added to Maple.1675Tue, 11 Oct 2005 00:00:00 ZDr. Janos PinterDr. Janos PinterChemical Equilibrium Model
http://www.maplesoft.com/applications/view.aspx?SID=1674&ref=Feed
Modeling chemical equilibrium of target compounds is of interest when controlling the pH, alkalinity or corrosivity of drinking water.
As part of this approach, one must determine the contribution rates of various components to mixtures which have given (known or prescribed) chemical characteristics.
In the example presented, we want to determine the concentrations of several components of phosphoric acid such that the resulting pH value is equal to 8, and the total phosphate concentration is 0.1 mols.
This worksheet requires that the Global Optimization Toolbox has been added to Maple.<img src="/view.aspx?si=1674/eqm.JPG" alt="Chemical Equilibrium Model" align="left"/>Modeling chemical equilibrium of target compounds is of interest when controlling the pH, alkalinity or corrosivity of drinking water.
As part of this approach, one must determine the contribution rates of various components to mixtures which have given (known or prescribed) chemical characteristics.
In the example presented, we want to determine the concentrations of several components of phosphoric acid such that the resulting pH value is equal to 8, and the total phosphate concentration is 0.1 mols.
This worksheet requires that the Global Optimization Toolbox has been added to Maple.1674Tue, 11 Oct 2005 00:00:00 ZDr. Janos PinterDr. Janos Pinter