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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.optim.linear;
import java.io.IOException;
import java.io.ObjectInputStream;
import java.io.ObjectOutputStream;
import java.io.Serializable;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.HashSet;
import java.util.List;
import java.util.Set;
import java.util.TreeSet;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.RealVector;
import org.apache.commons.math3.optim.nonlinear.scalar.GoalType;
import org.apache.commons.math3.optim.PointValuePair;
import org.apache.commons.math3.util.Precision;
/**
* A tableau for use in the Simplex method.
*
*
* Example:
*
* W | Z | x1 | x2 | x- | s1 | s2 | a1 | RHS
* ---------------------------------------------------
* -1 0 0 0 0 0 0 1 0 <= phase 1 objective
* 0 1 -15 -10 0 0 0 0 0 <= phase 2 objective
* 0 0 1 0 0 1 0 0 2 <= constraint 1
* 0 0 0 1 0 0 1 0 3 <= constraint 2
* 0 0 1 1 0 0 0 1 4 <= constraint 3
*
* W: Phase 1 objective function
* Z: Phase 2 objective function
* x1 & x2: Decision variables
* x-: Extra decision variable to allow for negative values
* s1 & s2: Slack/Surplus variables
* a1: Artificial variable
* RHS: Right hand side
*
* @since 2.0
*/
class SimplexTableau implements Serializable {
/** Column label for negative vars. */
private static final String NEGATIVE_VAR_COLUMN_LABEL = "x-";
/** Serializable version identifier. */
private static final long serialVersionUID = -1369660067587938365L;
/** Linear objective function. */
private final LinearObjectiveFunction f;
/** Linear constraints. */
private final List constraints;
/** Whether to restrict the variables to non-negative values. */
private final boolean restrictToNonNegative;
/** The variables each column represents */
private final List columnLabels = new ArrayList();
/** Simple tableau. */
private transient Array2DRowRealMatrix tableau;
/** Number of decision variables. */
private final int numDecisionVariables;
/** Number of slack variables. */
private final int numSlackVariables;
/** Number of artificial variables. */
private int numArtificialVariables;
/** Amount of error to accept when checking for optimality. */
private final double epsilon;
/** Amount of error to accept in floating point comparisons. */
private final int maxUlps;
/** Maps basic variables to row they are basic in. */
private int[] basicVariables;
/** Maps rows to their corresponding basic variables. */
private int[] basicRows;
/**
* Builds a tableau for a linear problem.
*
* @param f Linear objective function.
* @param constraints Linear constraints.
* @param goalType Optimization goal: either {@link GoalType#MAXIMIZE}
* or {@link GoalType#MINIMIZE}.
* @param restrictToNonNegative Whether to restrict the variables to non-negative values.
* @param epsilon Amount of error to accept when checking for optimality.
*/
SimplexTableau(final LinearObjectiveFunction f,
final Collection constraints,
final GoalType goalType,
final boolean restrictToNonNegative,
final double epsilon) {
this(f, constraints, goalType, restrictToNonNegative, epsilon, SimplexSolver.DEFAULT_ULPS);
}
/**
* Build a tableau for a linear problem.
* @param f linear objective function
* @param constraints linear constraints
* @param goalType type of optimization goal: either {@link GoalType#MAXIMIZE} or {@link GoalType#MINIMIZE}
* @param restrictToNonNegative whether to restrict the variables to non-negative values
* @param epsilon amount of error to accept when checking for optimality
* @param maxUlps amount of error to accept in floating point comparisons
*/
SimplexTableau(final LinearObjectiveFunction f,
final Collection constraints,
final GoalType goalType,
final boolean restrictToNonNegative,
final double epsilon,
final int maxUlps) {
this.f = f;
this.constraints = normalizeConstraints(constraints);
this.restrictToNonNegative = restrictToNonNegative;
this.epsilon = epsilon;
this.maxUlps = maxUlps;
this.numDecisionVariables = f.getCoefficients().getDimension() + (restrictToNonNegative ? 0 : 1);
this.numSlackVariables = getConstraintTypeCounts(Relationship.LEQ) +
getConstraintTypeCounts(Relationship.GEQ);
this.numArtificialVariables = getConstraintTypeCounts(Relationship.EQ) +
getConstraintTypeCounts(Relationship.GEQ);
this.tableau = createTableau(goalType == GoalType.MAXIMIZE);
// initialize the basic variables for phase 1:
// we know that only slack or artificial variables can be basic
initializeBasicVariables(getSlackVariableOffset());
initializeColumnLabels();
}
/**
* Initialize the labels for the columns.
*/
protected void initializeColumnLabels() {
if (getNumObjectiveFunctions() == 2) {
columnLabels.add("W");
}
columnLabels.add("Z");
for (int i = 0; i < getOriginalNumDecisionVariables(); i++) {
columnLabels.add("x" + i);
}
if (!restrictToNonNegative) {
columnLabels.add(NEGATIVE_VAR_COLUMN_LABEL);
}
for (int i = 0; i < getNumSlackVariables(); i++) {
columnLabels.add("s" + i);
}
for (int i = 0; i < getNumArtificialVariables(); i++) {
columnLabels.add("a" + i);
}
columnLabels.add("RHS");
}
/**
* Create the tableau by itself.
* @param maximize if true, goal is to maximize the objective function
* @return created tableau
*/
protected Array2DRowRealMatrix createTableau(final boolean maximize) {
// create a matrix of the correct size
int width = numDecisionVariables + numSlackVariables +
numArtificialVariables + getNumObjectiveFunctions() + 1; // + 1 is for RHS
int height = constraints.size() + getNumObjectiveFunctions();
Array2DRowRealMatrix matrix = new Array2DRowRealMatrix(height, width);
// initialize the objective function rows
if (getNumObjectiveFunctions() == 2) {
matrix.setEntry(0, 0, -1);
}
int zIndex = (getNumObjectiveFunctions() == 1) ? 0 : 1;
matrix.setEntry(zIndex, zIndex, maximize ? 1 : -1);
RealVector objectiveCoefficients = maximize ? f.getCoefficients().mapMultiply(-1) : f.getCoefficients();
copyArray(objectiveCoefficients.toArray(), matrix.getDataRef()[zIndex]);
matrix.setEntry(zIndex, width - 1, maximize ? f.getConstantTerm() : -1 * f.getConstantTerm());
if (!restrictToNonNegative) {
matrix.setEntry(zIndex, getSlackVariableOffset() - 1,
getInvertedCoefficientSum(objectiveCoefficients));
}
// initialize the constraint rows
int slackVar = 0;
int artificialVar = 0;
for (int i = 0; i < constraints.size(); i++) {
LinearConstraint constraint = constraints.get(i);
int row = getNumObjectiveFunctions() + i;
// decision variable coefficients
copyArray(constraint.getCoefficients().toArray(), matrix.getDataRef()[row]);
// x-
if (!restrictToNonNegative) {
matrix.setEntry(row, getSlackVariableOffset() - 1,
getInvertedCoefficientSum(constraint.getCoefficients()));
}
// RHS
matrix.setEntry(row, width - 1, constraint.getValue());
// slack variables
if (constraint.getRelationship() == Relationship.LEQ) {
matrix.setEntry(row, getSlackVariableOffset() + slackVar++, 1); // slack
} else if (constraint.getRelationship() == Relationship.GEQ) {
matrix.setEntry(row, getSlackVariableOffset() + slackVar++, -1); // excess
}
// artificial variables
if ((constraint.getRelationship() == Relationship.EQ) ||
(constraint.getRelationship() == Relationship.GEQ)) {
matrix.setEntry(0, getArtificialVariableOffset() + artificialVar, 1);
matrix.setEntry(row, getArtificialVariableOffset() + artificialVar++, 1);
matrix.setRowVector(0, matrix.getRowVector(0).subtract(matrix.getRowVector(row)));
}
}
return matrix;
}
/**
* Get new versions of the constraints which have positive right hand sides.
* @param originalConstraints original (not normalized) constraints
* @return new versions of the constraints
*/
public List normalizeConstraints(Collection originalConstraints) {
List normalized = new ArrayList(originalConstraints.size());
for (LinearConstraint constraint : originalConstraints) {
normalized.add(normalize(constraint));
}
return normalized;
}
/**
* Get a new equation equivalent to this one with a positive right hand side.
* @param constraint reference constraint
* @return new equation
*/
private LinearConstraint normalize(final LinearConstraint constraint) {
if (constraint.getValue() < 0) {
return new LinearConstraint(constraint.getCoefficients().mapMultiply(-1),
constraint.getRelationship().oppositeRelationship(),
-1 * constraint.getValue());
}
return new LinearConstraint(constraint.getCoefficients(),
constraint.getRelationship(), constraint.getValue());
}
/**
* Get the number of objective functions in this tableau.
* @return 2 for Phase 1. 1 for Phase 2.
*/
protected final int getNumObjectiveFunctions() {
return this.numArtificialVariables > 0 ? 2 : 1;
}
/**
* Get a count of constraints corresponding to a specified relationship.
* @param relationship relationship to count
* @return number of constraint with the specified relationship
*/
private int getConstraintTypeCounts(final Relationship relationship) {
int count = 0;
for (final LinearConstraint constraint : constraints) {
if (constraint.getRelationship() == relationship) {
++count;
}
}
return count;
}
/**
* Get the -1 times the sum of all coefficients in the given array.
* @param coefficients coefficients to sum
* @return the -1 times the sum of all coefficients in the given array.
*/
protected static double getInvertedCoefficientSum(final RealVector coefficients) {
double sum = 0;
for (double coefficient : coefficients.toArray()) {
sum -= coefficient;
}
return sum;
}
/**
* Checks whether the given column is basic.
* @param col index of the column to check
* @return the row that the variable is basic in. null if the column is not basic
*/
protected Integer getBasicRow(final int col) {
final int row = basicVariables[col];
return row == -1 ? null : row;
}
/**
* Returns the variable that is basic in this row.
* @param row the index of the row to check
* @return the variable that is basic for this row.
*/
protected int getBasicVariable(final int row) {
return basicRows[row];
}
/**
* Initializes the basic variable / row mapping.
* @param startColumn the column to start
*/
private void initializeBasicVariables(final int startColumn) {
basicVariables = new int[getWidth() - 1];
basicRows = new int[getHeight()];
Arrays.fill(basicVariables, -1);
for (int i = startColumn; i < getWidth() - 1; i++) {
Integer row = findBasicRow(i);
if (row != null) {
basicVariables[i] = row;
basicRows[row] = i;
}
}
}
/**
* Returns the row in which the given column is basic.
* @param col index of the column
* @return the row that the variable is basic in, or {@code null} if the variable is not basic.
*/
private Integer findBasicRow(final int col) {
Integer row = null;
for (int i = 0; i < getHeight(); i++) {
final double entry = getEntry(i, col);
if (Precision.equals(entry, 1d, maxUlps) && (row == null)) {
row = i;
} else if (!Precision.equals(entry, 0d, maxUlps)) {
return null;
}
}
return row;
}
/**
* Removes the phase 1 objective function, positive cost non-artificial variables,
* and the non-basic artificial variables from this tableau.
*/
protected void dropPhase1Objective() {
if (getNumObjectiveFunctions() == 1) {
return;
}
final Set columnsToDrop = new TreeSet();
columnsToDrop.add(0);
// positive cost non-artificial variables
for (int i = getNumObjectiveFunctions(); i < getArtificialVariableOffset(); i++) {
final double entry = getEntry(0, i);
if (Precision.compareTo(entry, 0d, epsilon) > 0) {
columnsToDrop.add(i);
}
}
// non-basic artificial variables
for (int i = 0; i < getNumArtificialVariables(); i++) {
int col = i + getArtificialVariableOffset();
if (getBasicRow(col) == null) {
columnsToDrop.add(col);
}
}
final double[][] matrix = new double[getHeight() - 1][getWidth() - columnsToDrop.size()];
for (int i = 1; i < getHeight(); i++) {
int col = 0;
for (int j = 0; j < getWidth(); j++) {
if (!columnsToDrop.contains(j)) {
matrix[i - 1][col++] = getEntry(i, j);
}
}
}
// remove the columns in reverse order so the indices are correct
Integer[] drop = columnsToDrop.toArray(new Integer[columnsToDrop.size()]);
for (int i = drop.length - 1; i >= 0; i--) {
columnLabels.remove((int) drop[i]);
}
this.tableau = new Array2DRowRealMatrix(matrix);
this.numArtificialVariables = 0;
// need to update the basic variable mappings as row/columns have been dropped
initializeBasicVariables(getNumObjectiveFunctions());
}
/**
* @param src the source array
* @param dest the destination array
*/
private void copyArray(final double[] src, final double[] dest) {
System.arraycopy(src, 0, dest, getNumObjectiveFunctions(), src.length);
}
/**
* Returns whether the problem is at an optimal state.
* @return whether the model has been solved
*/
boolean isOptimal() {
final double[] objectiveFunctionRow = getRow(0);
final int end = getRhsOffset();
for (int i = getNumObjectiveFunctions(); i < end; i++) {
final double entry = objectiveFunctionRow[i];
if (Precision.compareTo(entry, 0d, epsilon) < 0) {
return false;
}
}
return true;
}
/**
* Get the current solution.
* @return current solution
*/
protected PointValuePair getSolution() {
int negativeVarColumn = columnLabels.indexOf(NEGATIVE_VAR_COLUMN_LABEL);
Integer negativeVarBasicRow = negativeVarColumn > 0 ? getBasicRow(negativeVarColumn) : null;
double mostNegative = negativeVarBasicRow == null ? 0 : getEntry(negativeVarBasicRow, getRhsOffset());
final Set usedBasicRows = new HashSet();
final double[] coefficients = new double[getOriginalNumDecisionVariables()];
for (int i = 0; i < coefficients.length; i++) {
int colIndex = columnLabels.indexOf("x" + i);
if (colIndex < 0) {
coefficients[i] = 0;
continue;
}
Integer basicRow = getBasicRow(colIndex);
if (basicRow != null && basicRow == 0) {
// if the basic row is found to be the objective function row
// set the coefficient to 0 -> this case handles unconstrained
// variables that are still part of the objective function
coefficients[i] = 0;
} else if (usedBasicRows.contains(basicRow)) {
// if multiple variables can take a given value
// then we choose the first and set the rest equal to 0
coefficients[i] = 0 - (restrictToNonNegative ? 0 : mostNegative);
} else {
usedBasicRows.add(basicRow);
coefficients[i] =
(basicRow == null ? 0 : getEntry(basicRow, getRhsOffset())) -
(restrictToNonNegative ? 0 : mostNegative);
}
}
return new PointValuePair(coefficients, f.value(coefficients));
}
/**
* Perform the row operations of the simplex algorithm with the selected
* pivot column and row.
* @param pivotCol the pivot column
* @param pivotRow the pivot row
*/
protected void performRowOperations(int pivotCol, int pivotRow) {
// set the pivot element to 1
final double pivotVal = getEntry(pivotRow, pivotCol);
divideRow(pivotRow, pivotVal);
// set the rest of the pivot column to 0
for (int i = 0; i < getHeight(); i++) {
if (i != pivotRow) {
final double multiplier = getEntry(i, pivotCol);
if (multiplier != 0.0) {
subtractRow(i, pivotRow, multiplier);
}
}
}
// update the basic variable mappings
final int previousBasicVariable = getBasicVariable(pivotRow);
basicVariables[previousBasicVariable] = -1;
basicVariables[pivotCol] = pivotRow;
basicRows[pivotRow] = pivotCol;
}
/**
* Divides one row by a given divisor.
*
* After application of this operation, the following will hold:
*
dividendRow = dividendRow / divisor
*
* @param dividendRowIndex index of the row
* @param divisor value of the divisor
*/
protected void divideRow(final int dividendRowIndex, final double divisor) {
final double[] dividendRow = getRow(dividendRowIndex);
for (int j = 0; j < getWidth(); j++) {
dividendRow[j] /= divisor;
}
}
/**
* Subtracts a multiple of one row from another.
*
* After application of this operation, the following will hold:
*
minuendRow = minuendRow - multiple * subtrahendRow
*
* @param minuendRowIndex row index
* @param subtrahendRowIndex row index
* @param multiplier multiplication factor
*/
protected void subtractRow(final int minuendRowIndex, final int subtrahendRowIndex, final double multiplier) {
final double[] minuendRow = getRow(minuendRowIndex);
final double[] subtrahendRow = getRow(subtrahendRowIndex);
for (int i = 0; i < getWidth(); i++) {
minuendRow[i] -= subtrahendRow[i] * multiplier;
}
}
/**
* Get the width of the tableau.
* @return width of the tableau
*/
protected final int getWidth() {
return tableau.getColumnDimension();
}
/**
* Get the height of the tableau.
* @return height of the tableau
*/
protected final int getHeight() {
return tableau.getRowDimension();
}
/**
* Get an entry of the tableau.
* @param row row index
* @param column column index
* @return entry at (row, column)
*/
protected final double getEntry(final int row, final int column) {
return tableau.getEntry(row, column);
}
/**
* Set an entry of the tableau.
* @param row row index
* @param column column index
* @param value for the entry
*/
protected final void setEntry(final int row, final int column, final double value) {
tableau.setEntry(row, column, value);
}
/**
* Get the offset of the first slack variable.
* @return offset of the first slack variable
*/
protected final int getSlackVariableOffset() {
return getNumObjectiveFunctions() + numDecisionVariables;
}
/**
* Get the offset of the first artificial variable.
* @return offset of the first artificial variable
*/
protected final int getArtificialVariableOffset() {
return getNumObjectiveFunctions() + numDecisionVariables + numSlackVariables;
}
/**
* Get the offset of the right hand side.
* @return offset of the right hand side
*/
protected final int getRhsOffset() {
return getWidth() - 1;
}
/**
* Get the number of decision variables.
*
* If variables are not restricted to positive values, this will include 1 extra decision variable to represent
* the absolute value of the most negative variable.
*
* @return number of decision variables
* @see #getOriginalNumDecisionVariables()
*/
protected final int getNumDecisionVariables() {
return numDecisionVariables;
}
/**
* Get the original number of decision variables.
* @return original number of decision variables
* @see #getNumDecisionVariables()
*/
protected final int getOriginalNumDecisionVariables() {
return f.getCoefficients().getDimension();
}
/**
* Get the number of slack variables.
* @return number of slack variables
*/
protected final int getNumSlackVariables() {
return numSlackVariables;
}
/**
* Get the number of artificial variables.
* @return number of artificial variables
*/
protected final int getNumArtificialVariables() {
return numArtificialVariables;
}
/**
* Get the row from the tableau.
* @param row the row index
* @return the reference to the underlying row data
*/
protected final double[] getRow(int row) {
return tableau.getDataRef()[row];
}
/**
* Get the tableau data.
* @return tableau data
*/
protected final double[][] getData() {
return tableau.getData();
}
@Override
public boolean equals(Object other) {
if (this == other) {
return true;
}
if (other instanceof SimplexTableau) {
SimplexTableau rhs = (SimplexTableau) other;
return (restrictToNonNegative == rhs.restrictToNonNegative) &&
(numDecisionVariables == rhs.numDecisionVariables) &&
(numSlackVariables == rhs.numSlackVariables) &&
(numArtificialVariables == rhs.numArtificialVariables) &&
(epsilon == rhs.epsilon) &&
(maxUlps == rhs.maxUlps) &&
f.equals(rhs.f) &&
constraints.equals(rhs.constraints) &&
tableau.equals(rhs.tableau);
}
return false;
}
@Override
public int hashCode() {
return Boolean.valueOf(restrictToNonNegative).hashCode() ^
numDecisionVariables ^
numSlackVariables ^
numArtificialVariables ^
Double.valueOf(epsilon).hashCode() ^
maxUlps ^
f.hashCode() ^
constraints.hashCode() ^
tableau.hashCode();
}
/**
* Serialize the instance.
* @param oos stream where object should be written
* @throws IOException if object cannot be written to stream
*/
private void writeObject(ObjectOutputStream oos)
throws IOException {
oos.defaultWriteObject();
MatrixUtils.serializeRealMatrix(tableau, oos);
}
/**
* Deserialize the instance.
* @param ois stream from which the object should be read
* @throws ClassNotFoundException if a class in the stream cannot be found
* @throws IOException if object cannot be read from the stream
*/
private void readObject(ObjectInputStream ois)
throws ClassNotFoundException, IOException {
ois.defaultReadObject();
MatrixUtils.deserializeRealMatrix(this, "tableau", ois);
}
}