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OR-Objects FAQ

1. General
    1.1 - What is OR-Objects?
    1.2 - How much does it cost?
    1.3 - Can I use the freeware version for commercial applications?
    1.4- Can I use the freeware version on a client-server based application?

2. Standard - How Do I ...
    2.1 - Generate random numbers from a probability distribution?
    2.2 - Find the mean and standard deviation of a vector?
    2.3 - Integrate a function numerically?
    2.4 - Solve an equation numerically?
    2.5 - Find the inverse of a matrix?
    2.6 - Solve simultaneous linear equations?
    2.7 - Find the shortest path in a graph?
    2.8 - Optimize using linear programming?
    2.9 - Use probability distributions to model arrivals?
    2.10 - Color the vertices and edges of a graph?
    2.11 - Solve a Traveling Salesman Problem?
    2.12 - Solve a Vehicle Routing Problem?
    2.13 - Estimate the population mean, median and proportion from a sample?
    2.14 - Perform correlation analysis?
    2.15 - Solve multiple regression problems?

1. General

1.1 What is OR-Objects?

OR-Objects is a library of Java^tm classes for developing Operations Research applications. The purpose of OR-Objects is to provide a foundation of reusable software to speed the development of OR applications and make them more reliable. OR-Objects includes data structures and algorithms for developing custom solutions as well as many classical algorithms.

1.2 How much does it cost?

The freeware version of OR-Objects costs nothing. You can get a single copy and use it for private or commercal applications. The freeware version can not be redistributed.Here is the license agreement.

1.3 Can I use the freeware version for commercial applications?

Yes, here is the license agreement.

1.4 Can I use the freeware version on a server based application?

Yes, a single copy installed on the server. Here is the license agreement.

2. How Do I ...

2.1 Generate random numbers from a probability distribution?

The continuous and discrete distribution classes are contained in the probability package drasys.or.prob. Here is an example that generates random numbers from a normal distribution with a mean of 2.0 and standard deviation of 3.0.

import drasys.or.prob.*;

import drasys.or.matrix.*;

class RandomNumbers

{

public static void

main(String[] args)

{

// Create the normal distribution object.

double std = 2;

double mean = 3;

DistributionI dist = new NormalDistribution(mean, std);

// Get a random scaler.
double scaler = dist.getRandomScaler();

// Get a random vector.
VectorI vector = dist.getRandomVector(10);

        // Get a random matrix
        MatrixI matrix = dist.getRandomMatrix(10, 10);
    }
}

2.2 Easily find the mean and standard deviation of a vector?

The vector classes are contained in the matrix package drasys.or.matrix. Here is an example that creates a vector and then computes the mean and standard deviation.

import drasys.or.matrix.*;

class MeanStd

{

public static void

main(String[] args)

{

double[] array = {1,2,3,4,5,6};

VectorI vector = new DenseVector(array);

double mean = vector.sum()/vector.size();

double variance = vector.sumOfSquaredDifferences(mean) / (vector.size()-1);

double std = Math.sqrt(variance);

}

2.3 Integrate a function numerically?

The integration classes are contained in the nonlinear package drasys.or.nonlinear. Here is an example that integrates e^x from 0.0 to 1.0.

import drasys.or.nonlinear.*;

class Integrate

{

// This class defines the function to integrate.

static class MyFunction

implements FunctionI

{

public double

function(double x)

{

return Math.exp(x);

}

public static void

main(String[] args)

{

// This example uses trapezoidal integration.

IntegrationI integration = new Trapezoidal();

try{

// Integrate MyFunction from 0.0 to 1.0.

double in = integration.integrate(new MyFunction(), 0.0, 1.0);

System.out.println(in);

}

catch(AccuracyException e){}

}

2.4 Solve an equation numerically?

The equation solution classes are contained in the nonlinear package drasys.or.nonlinear. Here is an example that solves for the value of x in gamma(x) = 4.0;

import drasys.or.*;

import drasys.or.nonlinear.*;

class Solve

{

// This class defines the function to solve.

static class MyFunction

implements FunctionI

{

public double

function(double x)

{

return Math.exp(new Functions().lnGamma(x));

}

public static void

main(String[] args)

{

// This example uses the bisection algorithm.

EquationSolutionI solution = new Bisection();

try{

// We expect x be bracketed by 3.0 and 4.0.

double lower = 3;

double upper = 4;

// Solve MyFunction for the x that results in 4.0.

double x = solution.solve(new MyFunction(), lower, upper, 4.0);

System.out.println(x);

}

catch(NonlinearException e){}

}

2.5 Find the inverse of a matrix?

The linear algebra classes are contained in the algebra package drasys.or.linear.algebra. Here is an example that inverts a random matrix and then checks the results.

import drasys.or.*;

import drasys.or.prob.*;

import drasys.or.matrix.*;

import drasys.or.linear.*;

import drasys.or.linear.algebra.*;

class Inverse

{

public static void

main(String[] args)

{

try{

DistributionI dist = new UniformDistribution(1.0, 2.0);

MatrixI matrix = dist.getRandomMatrix(10, 10);

// Compute the inverse using the default algebraic operations class

AlgebraI algebra = new Algebra();

MatrixI inverse = algebra.invert(matrix);

               // A matrix multiplied by its inverse should equal the identity matrix
            MatrixI test = algebra.multiply(inverse, matrix);
            MatrixI identity = new SparseMatrix(new DenseVector(10, 1.0));
            if(test.equals(identity))
                System.out.println("OK");
            else
               System.out.println("ERROR");
        }
        catch(LinearException e){}
    }
}

2.6 Solve simultaneous linear equations?

The decomposition classes are contained in the algebra package drasys.or.linear.algabra. Here is an example that solves a set of simultaneous equations for multiple right hand side vectors.

import drasys.or.*;

import drasys.or.prob.*;

import drasys.or.matrix.*;

import drasys.or.linear.*;

import drasys.or.linear.algebra.*;

class Simultaneous
{
    public static void
    main(String[] args)
    {
         try{
             AlgebraI algebra = new Algebra();
             DistributionI dist = new UniformDistribution(1.0, 2.0);
           DecompositionI decomp = new QRIteration();

           // Build and decompose the equations.
             MatrixI equations = dist.getRandomMatrix(10, 10);
             decomp.decompose(equations);

           // Solve first set
             VectorI rightHandSide1 = dist.getRandomVector(10);
             VectorI solution1 = decomp.solveEquations(rightHandSide1);

           // The equations multiplied by the solution should equal the rhs.
             if(algebra.multiply(equations, solution1).equals(rightHandSide1))
                System.out.println("OK-1");
             else
                System.out.println("ERROR-1");

           // Solve second set
             VectorI rightHandSide2 = dist.getRandomVector(10);
             VectorI solution2 = decomp.solveEquations(rightHandSide2);

           // The equations multiplied by the solution should equal the rhs.
             if(algebra.multiply(equations, solution2).equals(rightHandSide2))
                System.out.println("OK-2");
             else
                System.out.println("ERROR-2");
         }
         catch(LinearException e){}
     }
}

2.7 Find the shortest path in a graph?

The graph classes are contained in the graph package drasys.or.graph. There are two tutorials in the OR-Objects Tutorials that demonstrate the use of shortest path algorithms. Tutorial-1 shows how to use OR-Objects to find the shortest path between two vertices. Tutorial-2 shows how to find multiple paths to or from a single vertex.

2.8 Optimize using linear programming?

The linear programming classes are contained in the mathematical programming package drasys.or.mp.lp. This example shows how to use OR-Objects to optimize a small linear model using the simplex algorithm.

Linear Model
`(max) Z = 20x₁ + 15x₂` `c1: x₁ <= 100` `c2: x₂ <= 100` `c3: 50x₁ + 35x₂ <= 6000` `c4: 20x₁ + 15x₂ >= 2000`

import drasys.or.mp.*;
import drasys.or.mp.lp.*;

public class MpExample
{

    public static void
    main(String[] args)
    throws Exception
    {
        SizableProblemI prob = new Problem(4,2);
        prob.getMetadata().put("lp.isMaximize", "true");

prob.newVariable("x1").setObjectiveCoefficient(20.0);
prob.newVariable("x2").setObjectiveCoefficient(15.0);

        final byte LTE = Constraint.LESS;
        final byte GTE = Constraint.GREATER;
        prob.newConstraint("c1").setType(LTE).setRightHandSide( 100.0);
        prob.newConstraint("c2").setType(LTE).setRightHandSide( 100.0);
        prob.newConstraint("c3").setType(LTE).setRightHandSide( 6000.0);
        prob.newConstraint("c4").setType(GTE).setRightHandSide( 2000.0);

        // Columnwise insertion is more efficient for the default matrix.
        prob.setCoefficientAt("c1", "x1", 1.0);
        prob.setCoefficientAt("c3", "x1", 50.0);
        prob.setCoefficientAt("c4", "x1", 20.0);
        prob.setCoefficientAt("c2", "x2", 1.0);
        prob.setCoefficientAt("c3", "x2", 35.0);
        prob.setCoefficientAt("c4", "x2", 15.0);

        LinearProgrammingI lp = new DenseSimplex(prob);
        double ans = lp.solve();
        System.out.println("Solution = "+ans);
    }

}

2.9 Use probability distributions to model arrivals?

The probability package drasys.or.prob contains the probability distribution classes and Tutorial-3 demonstrates the use of the ExponentialDistribution and the EmpiricalDistribution to model passengers entering an elevator system.

2.10 Color the vertices and edges of a graph?

The graph classes are contained in the graph coloring package drasys.or.graph.color. This examples shows how to use OR-Objects to color the vertices and edges of the following graph where ths vertices are labeled A through G and the edges are labeled one through ten.

import java.util.*;

import drasys.or.graph.*;

import drasys.or.graph.color.*;

class Coloring
{
    public static void
    main(String[] args)
    {
        String[] colors = {"red", "green", "blue", "grey"};
        SparseGraph graph = new SparseGraph();

        try{
        // Add Vertices
        graph.addVertex("A"); graph.addVertex("B");
        graph.addVertex("C"); graph.addVertex("D");
        graph.addVertex("E"); graph.addVertex("F");
        graph.addVertex("G");

        // Add Edges
        graph.addEdge("A","B","2"); graph.addEdge("A","C","1");
        graph.addEdge("D","B","3"); graph.addEdge("D","C","7");
        graph.addEdge("D","E","5"); graph.addEdge("D","F","9");
        graph.addEdge("G","E","6"); graph.addEdge("G","F","10");
        graph.addEdge("E","B","4",true);
        graph.addEdge("C","F","8",true);

        // Create algorithm and color elements
        ColoringI alg = new WelshPowell(graph);
        alg.colorEdges();
        alg.colorVertices();

        // Print Vertex Colors
        System.out.println("\n--- Vertex Colors ---");
        for(Enumeration enum = graph.vertices(); enum.hasMoreElements();){
            VertexI vert = (VertexI)enum.nextElement();
            String label = (String)vert.getKey();
            String color = colors[alg.getVertexColor(vert)];
            System.out.println(label + " = " + color);
        }

        // Print Edge Colors
        System.out.println("\n--- Edge Colors ---");
        for(Enumeration enum = graph.edges(); enum.hasMoreElements();){
            EdgeI edge = (EdgeI)enum.nextElement();
            String label = (String)edge.getValue();
            String color = colors[alg.getEdgeColor(edge)];
            System.out.println(label + " = " + color);
        }

}catch(Exception e){}
}
}

2.11 Solve a Traveling Salesman Problem?

The TSP package drasys.or.graph.tsp contains the traveling salesman problem classes and Tutorial-4 demonstrates the use of several of the TSP algorithms in OR-Objects.

2.12 Solve a Vehicle Routing Problem?

The VRP package drasys.or.graph.vrp contains the traveling salesman problem classes and Tutorial-6 demonstrates the use of several of the VRP algorithms in OR-Objects.

2.13 Estimate the population mean, median and proportion from a sample?

The statistical estimation classes are contained in the package drasys.or.stat.est.Here is an example that creates a sample and then estimates the mean, median and proportion along with the associated confidence intervals.

import drasys.or.matrix.*;

import drasys.or.stat.est.*;

class Estimate

{

public static void

main(String[] args)

{

double[] array = {1,2,3,4,5,6};

VectorI sample = new DenseVector(array);

        PopulationMean pMean = new PopulationMean(sample);
        double mean = pMean.getMean();
        double meanLowerBound = pMean.getLowerBound(0.95);
        double meanUpperBound = pMean.getUpperBound(0.95);

        PopulationMedian pMedian = new PopulationMedian(sample);
        double median = pMean.getMedian();
        double medianLowerBound = pMedian.getLowerBound(0.95);
        double medianUpperBound = pMedian.getUpperBound(0.95);

        PopulationProportion pProp = new PopulationProportion(sample, 4);
        double prop = pMean.getProportion();
        double propLowerBound = pProp.getLowerBound(0.95);
        double propUpperBound = pProp.getUpperBound(0.95);
    }
}

3. Professional - How Do I ...

2.14 Perform correlation analysis?

        double[][]data = {
            {47,4.2}, {71,8.1}, {64,6.8}, {35,4.3}, {43,5.0},
            {60,7.5}, {38,4.7}, {59,5.9}, {67,6.9}, {56,5.7},
            {67,5.7}, {57,5.4}, {69,7.5}, {38,3.8}, {54,5.9},
            {76,6.3}, {53,5.7}, {40,4.0}, {47,5.2}, {23,2.2},
        };

LinearCorrelation lc = new LinearCorrelation(new ColumnMajorMatrix(data));

        MatrixI sumSquares = lc.getSumOfSquares();
        System.out.println("Sum of squares for col-0 = " + sumSquares.elementAt(0);
        System.out.println("Sum of squares for col-1 = " + sumSquares.elementAt(1);

        MatrixI sumProd = lc.getSumOfProducts();
        System.out.println("Sum of squares for col-0 = " + sumProd.elementAt(0,0);
        System.out.println("Sum of squares for col-1 = " + sumProd.elementAt(1,1);
        System.out.println("Sum of products for col-0 and col-1 = " + sumProd.elementAt(0,1);

MatrixI correlation = lc.getCorrelation();
System.out.println("Correlation for col-0 and col-1 = " + correlation.elementAt(0,1);

MatrixI upperBound = lc.getUpperBound(0.95);
System.out.println("Correlation upper bound for col-0 and col-1 = " + upperBound.elementAt(0,1);

MatrixI lowerBound = lc.getLowerBound(0.95);
System.out.println("Correlation lower bound for col-0 and col-1 = " + lowerBound.elementAt(0,1);

2.15 Solve multiple regression problems?

        double[] dep = {
            19929, 5839, 23696, 9881, 30011,
             7241, 11634, 45684, 36476, 12068,
        };

        double[][] indep = {
            {10.5, 2200}, { 2.5, 1000}, {13.1, 3250}, { 4.0, 1475}, {14.7, 3800},
            { 3.6, 1200}, { 7.1, 1900}, {22.5, 5575}, {17.0, 4200}, { 6.4, 1850},
        };

        VectorI dependent = new DenseVector(da);
        MatrixI independent = new RowArrayMatrix(ia, true);
        LinearRegressionI reg = new ReverseLinear(dependent, independent);

        // Print coefficients
        VectorI coef = reg.getCoefficients();
        System.out.println("Intercept Coefficient = "+coef.elementAt(2));
        System.out.println("Column-0 Coefficient = "+coef.elementAt(0));
        System.out.println("Column-1 Coefficient = "+coef.elementAt(1));

//Print regression report
System.out.println(reg);