    001    /*
    002     * Licensed to the Apache Software Foundation (ASF) under one or more
    003     * contributor license agreements.  See the NOTICE file distributed with
    004     * this work for additional information regarding copyright ownership.
    005     * The ASF licenses this file to You under the Apache License, Version 2.0
    006     * (the "License"); you may not use this file except in compliance with
    007     * the License.  You may obtain a copy of the License at
    008     *
    009     *      http://www.apache.org/licenses/LICENSE-2.0
    010     *
    011     * Unless required by applicable law or agreed to in writing, software
    012     * distributed under the License is distributed on an "AS IS" BASIS,
    013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    014     * See the License for the specific language governing permissions and
    015     * limitations under the License.
    016     */
    017    package org.apache.commons.math3.optim.linear;
    018
    019    import java.util.ArrayList;
    020    import java.util.List;
    021    import org.apache.commons.math3.exception.TooManyIterationsException;
    022    import org.apache.commons.math3.optim.PointValuePair;
    023    import org.apache.commons.math3.util.Precision;
    024
    025    /**
    026     * Solves a linear problem using the "Two-Phase Simplex" method.
    027     *
    028     * @version $Id: SimplexSolver.java 1416643 2012-12-03 19:37:14Z tn $
    029     * @since 2.0
    030     */
    031    public class SimplexSolver extends LinearOptimizer {
    032        /** Default amount of error to accept for algorithm convergence. */
    033        private static final double DEFAULT_EPSILON = 1.0e-6;
    034
    035        /** Default amount of error to accept in floating point comparisons (as ulps). */
    036        private static final int DEFAULT_ULPS = 10;
    037
    038        /** Amount of error to accept for algorithm convergence. */
    039        private final double epsilon;
    040
    041        /** Amount of error to accept in floating point comparisons (as ulps). */
    042        private final int maxUlps;
    043
    044        /**
    045         * Builds a simplex solver with default settings.
    046         */
    047        public SimplexSolver() {
    048            this(DEFAULT_EPSILON, DEFAULT_ULPS);
    049        }
    050
    051        /**
    052         * Builds a simplex solver with a specified accepted amount of error.
    053         *
    054         * @param epsilon Amount of error to accept for algorithm convergence.
    055         * @param maxUlps Amount of error to accept in floating point comparisons.
    056         */
    057        public SimplexSolver(final double epsilon,
    058                             final int maxUlps) {
    059            this.epsilon = epsilon;
    060            this.maxUlps = maxUlps;
    061        }
    062
    063        /**
    064         * Returns the column with the most negative coefficient in the objective function row.
    065         *
    066         * @param tableau Simple tableau for the problem.
    067         * @return the column with the most negative coefficient.
    068         */
    069        private Integer getPivotColumn(SimplexTableau tableau) {
    070            double minValue = 0;
    071            Integer minPos = null;
    072            for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getWidth() - 1; i++) {
    073                final double entry = tableau.getEntry(0, i);
    074                // check if the entry is strictly smaller than the current minimum
    075                // do not use a ulp/epsilon check
    076                if (entry < minValue) {
    077                    minValue = entry;
    078                    minPos = i;
    079                }
    080            }
    081            return minPos;
    082        }
    083
    084        /**
    085         * Returns the row with the minimum ratio as given by the minimum ratio test (MRT).
    086         *
    087         * @param tableau Simple tableau for the problem.
    088         * @param col Column to test the ratio of (see {@link #getPivotColumn(SimplexTableau)}).
    089         * @return the row with the minimum ratio.
    090         */
    091        private Integer getPivotRow(SimplexTableau tableau, final int col) {
    092            // create a list of all the rows that tie for the lowest score in the minimum ratio test
    093            List<Integer> minRatioPositions = new ArrayList<Integer>();
    094            double minRatio = Double.MAX_VALUE;
    095            for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) {
    096                final double rhs = tableau.getEntry(i, tableau.getWidth() - 1);
    097                final double entry = tableau.getEntry(i, col);
    098
    099                if (Precision.compareTo(entry, 0d, maxUlps) > 0) {
    100                    final double ratio = rhs / entry;
    101                    // check if the entry is strictly equal to the current min ratio
    102                    // do not use a ulp/epsilon check
    103                    final int cmp = Double.compare(ratio, minRatio);
    104                    if (cmp == 0) {
    105                        minRatioPositions.add(i);
    106                    } else if (cmp < 0) {
    107                        minRatio = ratio;
    108                        minRatioPositions = new ArrayList<Integer>();
    109                        minRatioPositions.add(i);
    110                    }
    111                }
    112            }
    113
    114            if (minRatioPositions.size() == 0) {
    115                return null;
    116            } else if (minRatioPositions.size() > 1) {
    117                // there's a degeneracy as indicated by a tie in the minimum ratio test
    118
    119                // 1. check if there's an artificial variable that can be forced out of the basis
    120                if (tableau.getNumArtificialVariables() > 0) {
    121                    for (Integer row : minRatioPositions) {
    122                        for (int i = 0; i < tableau.getNumArtificialVariables(); i++) {
    123                            int column = i + tableau.getArtificialVariableOffset();
    124                            final double entry = tableau.getEntry(row, column);
    125                            if (Precision.equals(entry, 1d, maxUlps) && row.equals(tableau.getBasicRow(column))) {
    126                                return row;
    127                            }
    128                        }
    129                    }
    130                }
    131
    132                // 2. apply Bland's rule to prevent cycling:
    133                //    take the row for which the corresponding basic variable has the smallest index
    134                //
    135                // see http://www.stanford.edu/class/msande310/blandrule.pdf
    136                // see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper)
    137                //
    138                // Additional heuristic: if we did not get a solution after half of maxIterations
    139                //                       revert to the simple case of just returning the top-most row
    140                // This heuristic is based on empirical data gathered while investigating MATH-828.
    141                if (getEvaluations() < getMaxEvaluations() / 2) {
    142                    Integer minRow = null;
    143                    int minIndex = tableau.getWidth();
    144                    final int varStart = tableau.getNumObjectiveFunctions();
    145                    final int varEnd = tableau.getWidth() - 1;
    146                    for (Integer row : minRatioPositions) {
    147                        for (int i = varStart; i < varEnd && !row.equals(minRow); i++) {
    148                            final Integer basicRow = tableau.getBasicRow(i);
    149                            if (basicRow != null && basicRow.equals(row)) {
    150                                if (i < minIndex) {
    151                                    minIndex = i;
    152                                    minRow = row;
    153                                }
    154                            }
    155                        }
    156                    }
    157                    return minRow;
    158                }
    159            }
    160            return minRatioPositions.get(0);
    161        }
    162
    163        /**
    164         * Runs one iteration of the Simplex method on the given model.
    165         *
    166         * @param tableau Simple tableau for the problem.
    167         * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
    168         * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
    169         */
    170        protected void doIteration(final SimplexTableau tableau)
    171            throws TooManyIterationsException,
    172                   UnboundedSolutionException {
    173
    174            incrementIterationCount();
    175
    176            Integer pivotCol = getPivotColumn(tableau);
    177            Integer pivotRow = getPivotRow(tableau, pivotCol);
    178            if (pivotRow == null) {
    179                throw new UnboundedSolutionException();
    180            }
    181
    182            // set the pivot element to 1
    183            double pivotVal = tableau.getEntry(pivotRow, pivotCol);
    184            tableau.divideRow(pivotRow, pivotVal);
    185
    186            // set the rest of the pivot column to 0
    187            for (int i = 0; i < tableau.getHeight(); i++) {
    188                if (i != pivotRow) {
    189                    final double multiplier = tableau.getEntry(i, pivotCol);
    190                    tableau.subtractRow(i, pivotRow, multiplier);
    191                }
    192            }
    193        }
    194
    195        /**
    196         * Solves Phase 1 of the Simplex method.
    197         *
    198         * @param tableau Simple tableau for the problem.
    199         * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
    200         * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
    201         * @throws NoFeasibleSolutionException if there is no feasible solution?
    202         */
    203        protected void solvePhase1(final SimplexTableau tableau)
    204            throws TooManyIterationsException,
    205                   UnboundedSolutionException,
    206                   NoFeasibleSolutionException {
    207
    208            // make sure we're in Phase 1
    209            if (tableau.getNumArtificialVariables() == 0) {
    210                return;
    211            }
    212
    213            while (!tableau.isOptimal()) {
    214                doIteration(tableau);
    215            }
    216
    217            // if W is not zero then we have no feasible solution
    218            if (!Precision.equals(tableau.getEntry(0, tableau.getRhsOffset()), 0d, epsilon)) {
    219                throw new NoFeasibleSolutionException();
    220            }
    221        }
    222
    223        /** {@inheritDoc} */
    224        @Override
    225        public PointValuePair doOptimize()
    226            throws TooManyIterationsException,
    227                   UnboundedSolutionException,
    228                   NoFeasibleSolutionException {
    229            final SimplexTableau tableau =
    230                new SimplexTableau(getFunction(),
    231                                   getConstraints(),
    232                                   getGoalType(),
    233                                   isRestrictedToNonNegative(),
    234                                   epsilon,
    235                                   maxUlps);
    236
    237            solvePhase1(tableau);
    238            tableau.dropPhase1Objective();
    239
    240            while (!tableau.isOptimal()) {
    241                doIteration(tableau);
    242            }
    243            return tableau.getSolution();
    244        }
    245    }