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#include "qsimplex_p.h"

#include <QtCore/qset.h>

#include <QtCore/qdebug.h>

#include <stdlib.h>

QT_BEGIN_NAMESPACE

/*!

  \internal

  \class QSimplex

  The QSimplex class is a Linear Programming problem solver based on the two-phase

  simplex method.

  It takes a set of QSimplexConstraints as its restrictive constraints and an

  additional QSimplexConstraint as its objective function. Then methods to maximize

  and minimize the problem solution are provided.

  The two-phase simplex method is based on the following steps:

  First phase:

  1.a) Modify the original, complex, and possibly not feasible problem, into a new,

       easy to solve problem.

  1.b) Set as the objective of the new problem, a feasible solution for the original

       complex problem.

  1.c) Run simplex to optimize the modified problem and check whether a solution for

       the original problem exists.

  Second phase:

  2.a) Go back to the original problem with the feasibl (but not optimal) solution

       found in the first phase.

  2.b) Set the original objective.

  3.c) Run simplex to optimize the original problem towards its optimal solution.

*/

/*!

  \internal

*/

QSimplex::QSimplex() : objective(0), rows(0), columns(0), firstArtificial(0), matrix(0)

{

}

/*!

  \internal

*/

QSimplex::~QSimplex()

{

    clearDataStructures();

}

/*!

  \internal

*/

void QSimplex::clearDataStructures()

{

    if (matrix == 0)

        return;

    // Matrix

    rows = 0;

    columns = 0;

    firstArtificial = 0;

    free(matrix);

    matrix = 0;

    // Constraints

    for (int i = 0; i < constraints.size(); ++i) {

        delete constraints[i]->helper.first;

        delete constraints[i]->artificial;

        delete constraints[i];

    }

    constraints.clear();

    // Other

    variables.clear();

    objective = 0;

}

/*!

  \internal

  Sets the new constraints in the simplex solver and returns whether the problem

  is feasible.

  This method sets the new constraints, normalizes them, creates the simplex matrix

  and runs the first simplex phase.

*/

bool QSimplex::setConstraints(const QList<QSimplexConstraint *> newConstraints)

{

    ////////////////////////////

    // Reset to initial state //

    ////////////////////////////

    clearDataStructures();

    if (newConstraints.isEmpty())

        return true;    // we are ok with no constraints

    // Make deep copy of constraints. We need this copy because we may change

    // them in the simplification method.

    for (int i = 0; i < newConstraints.size(); ++i) {

        QSimplexConstraint *c = new QSimplexConstraint;

        c->constant = newConstraints[i]->constant;

        c->ratio = newConstraints[i]->ratio;

        c->variables = newConstraints[i]->variables;

        constraints << c;

    }

    // Remove constraints of type Var == K and replace them for their value.

    if (!simplifyConstraints(&constraints)) {

        qWarning() << "QSimplex: No feasible solution!";

        clearDataStructures();

        return false;

    }

    ///////////////////////////////////////

    // Prepare variables and constraints //

    ///////////////////////////////////////

    // Set Variables direct mapping.

    // "variables" is a list that provides a stable, indexed list of all variables

    // used in this problem.

    QSet<QSimplexVariable *> variablesSet;

    for (int i = 0; i < constraints.size(); ++i)

        variablesSet += \

            QSet<QSimplexVariable *>::fromList(constraints[i]->variables.keys());

    variables = variablesSet.toList();

    // Set Variables reverse mapping

    // We also need to be able to find the index for a given variable, to do that

    // we store in each variable its index.

    for (int i = 0; i < variables.size(); ++i) {

        // The variable "0" goes at the column "1", etc...

        variables[i]->index = i + 1;

    }

    // Normalize Constraints

    // In this step, we prepare the constraints in two ways:

    // Firstly, we modify all constraints of type "LessOrEqual" or "MoreOrEqual"

    // by the adding slack or surplus variables and making them "Equal" constraints.

    // Secondly, we need every single constraint to have a direct, easy feasible

    // solution. Constraints that have slack variables are already easy to solve,

    // to all the others we add artificial variables.

    //

    // At the end we modify the constraints as follows:

    //  - LessOrEqual: SLACK variable is added.

    //  - Equal: ARTIFICIAL variable is added.

    //  - More or Equal: ARTIFICIAL and SURPLUS variables are added.

    int variableIndex = variables.size();

    QList <QSimplexVariable *> artificialList;

    for (int i = 0; i < constraints.size(); ++i) {

        QSimplexVariable *slack;

        QSimplexVariable *surplus;

        QSimplexVariable *artificial;

        Q_ASSERT(constraints[i]->helper.first == 0);

        Q_ASSERT(constraints[i]->artificial == 0);

        switch(constraints[i]->ratio) {

        case QSimplexConstraint::LessOrEqual:

            slack = new QSimplexVariable;

            slack->index = ++variableIndex;

            constraints[i]->helper.first = slack;

            constraints[i]->helper.second = 1.0;

            break;

        case QSimplexConstraint::MoreOrEqual:

            surplus = new QSimplexVariable;

            surplus->index = ++variableIndex;

            constraints[i]->helper.first = surplus;

            constraints[i]->helper.second = -1.0;

            // fall through

        case QSimplexConstraint::Equal:

            artificial = new QSimplexVariable;

            constraints[i]->artificial = artificial;

            artificialList += constraints[i]->artificial;

            break;

        }

    }

    // All original, slack and surplus have already had its index set

    // at this point. We now set the index of the artificial variables

    // as to ensure they are at the end of the variable list and therefore

    // can be easily removed at the end of this method.

    firstArtificial = variableIndex + 1;

    for (int i = 0; i < artificialList.size(); ++i)

        artificialList[i]->index = ++variableIndex;

    artificialList.clear();

    /////////////////////////////

    // Fill the Simplex matrix //

    /////////////////////////////

    // One for each variable plus the Basic and BFS columns (first and last)

    columns = variableIndex + 2;

    // One for each constraint plus the objective function

    rows = constraints.size() + 1;

    matrix = (qreal *)malloc(sizeof(qreal) * columns * rows);

    if (!matrix) {

        qWarning() << "QSimplex: Unable to allocate memory!";

        return false;

    }

    for (int i = columns * rows - 1; i >= 0; --i)

        matrix[i] = 0.0;

    // Fill Matrix

    for (int i = 1; i <= constraints.size(); ++i) {

        QSimplexConstraint *c = constraints[i - 1];

        if (c->artificial) {

            // Will use artificial basic variable

            setValueAt(i, 0, c->artificial->index);

            setValueAt(i, c->artificial->index, 1.0);

            if (c->helper.second != 0.0) {

                // Surplus variable

                setValueAt(i, c->helper.first->index, c->helper.second);

            }

        } else {

            // Slack is used as the basic variable

            Q_ASSERT(c->helper.second == 1.0);

            setValueAt(i, 0, c->helper.first->index);

            setValueAt(i, c->helper.first->index, 1.0);

        }

        QHash<QSimplexVariable *, qreal>::const_iterator iter;

        for (iter = c->variables.constBegin();

             iter != c->variables.constEnd();

             ++iter) {

            setValueAt(i, iter.key()->index, iter.value());

        }

        setValueAt(i, columns - 1, c->constant);

    }

    // Set objective for the first-phase Simplex.

    // Z = -1 * sum_of_artificial_vars

    for (int j = firstArtificial; j < columns - 1; ++j)

        setValueAt(0, j, 1.0);

    // Maximize our objective (artificial vars go to zero)

    solveMaxHelper();

    // If there is a solution where the sum of all artificial

    // variables is zero, then all of them can be removed and yet

    // we will have a feasible (but not optimal) solution for the

    // original problem.

    // Otherwise, we clean up our structures and report there is

    // no feasible solution.

    if ((valueAt(0, columns - 1) != 0.0) && (qAbs(valueAt(0, columns - 1)) > 0.00001)) {

        qWarning() << "QSimplex: No feasible solution!";

        clearDataStructures();

        return false;

    }

    // Remove artificial variables. We already have a feasible

    // solution for the first problem, thus we don't need them

    // anymore.

    clearColumns(firstArtificial, columns - 2);

    return true;

}

/*!

  \internal

  Run simplex on the current matrix with the current objective.

  This is the iterative method. The matrix lines are combined

  as to modify the variable values towards the best solution possible.

  The method returns when the matrix is in the optimal state.

*/

void QSimplex::solveMaxHelper()

{

    reducedRowEchelon();

    while (iterate()) ;

}

/*!

  \internal

*/

void QSimplex::setObjective(QSimplexConstraint *newObjective)

{

    objective = newObjective;

}

/*!

  \internal

*/

void QSimplex::clearRow(int rowIndex)

{

    qreal *item = matrix + rowIndex * columns;

    for (int i = 0; i < columns; ++i)

        item[i] = 0.0;

}

/*!

  \internal

*/

void QSimplex::clearColumns(int first, int last)

{

    for (int i = 0; i < rows; ++i) {

        qreal *row = matrix + i * columns;

        for (int j = first; j <= last; ++j)

            row[j] = 0.0;

    }

}

/*!

  \internal

*/

void QSimplex::dumpMatrix()

{

    qDebug("---- Simplex Matrix ----\n");

    QString str(QLatin1String("       "));

    for (int j = 0; j < columns; ++j)

        str += QString::fromAscii("  <%1 >").arg(j, 2);

    qDebug("%s", qPrintable(str));

    for (int i = 0; i < rows; ++i) {

        str = QString::fromAscii("Row %1:").arg(i, 2);

        qreal *row = matrix + i * columns;

        for (int j = 0; j < columns; ++j)

            str += QString::fromAscii("%1").arg(row[j], 7, 'f', 2);

        qDebug("%s", qPrintable(str));

    }

    qDebug("------------------------\n");

}

/*!

  \internal

*/

void QSimplex::combineRows(int toIndex, int fromIndex, qreal factor)

{

    if (!factor)

        return;

    qreal *from = matrix + fromIndex * columns;

    qreal *to = matrix + toIndex * columns;

    for (int j = 1; j < columns; ++j) {

        qreal value = from[j];

        // skip to[j] = to[j] + factor*0.0

        if (value == 0.0)

            continue;

        to[j] += factor * value;

        // ### Avoid Numerical errors

        if (qAbs(to[j]) < 0.0000000001)

            to[j] = 0.0;

    }

}

/*!

  \internal

*/

int QSimplex::findPivotColumn()

{

    qreal min = 0;

    int minIndex = -1;

    for (int j = 0; j < columns-1; ++j) {

        if (valueAt(0, j) < min) {

            min = valueAt(0, j);

            minIndex = j;

        }

    }

    return minIndex;

}

/*!

  \internal

  For a given pivot column, find the pivot row. That is, the row with the

  minimum associated "quotient" where:

  - quotient is the division of the value in the last column by the value

    in the pivot column.

  - rows with value less or equal to zero are ignored

  - if two rows have the same quotient, lines are chosen based on the

    highest variable index (value in the first column)

  The last condition avoids a bug where artificial variables would be

  left behind for the second-phase simplex, and with 'good'

  constraints would be removed before it, what would lead to incorrect

  results.

*/

int QSimplex::pivotRowForColumn(int column)

{

    qreal min = qreal(999999999999.0); // ###

    int minIndex = -1;

    for (int i = 1; i < rows; ++i) {

        qreal divisor = valueAt(i, column);

        if (divisor <= 0)

            continue;

        qreal quotient = valueAt(i, columns - 1) / divisor;

        if (quotient < min) {

            min = quotient;

            minIndex = i;

        } else if ((quotient == min) && (valueAt(i, 0) > valueAt(minIndex, 0))) {

            minIndex = i;

        }

    }

    return minIndex;

}

/*!

  \internal

*/

void QSimplex::reducedRowEchelon()

{

    for (int i = 1; i < rows; ++i) {

        int factorInObjectiveRow = valueAt(i, 0);

        combineRows(0, i, -1 * valueAt(0, factorInObjectiveRow));

    }

}

/*!

  \internal

  Does one iteration towards a better solution for the problem.

  See 'solveMaxHelper'.

*/

bool QSimplex::iterate()

{

    // Find Pivot column

    int pivotColumn = findPivotColumn();

    if (pivotColumn == -1)

        return false;

    // Find Pivot row for column

    int pivotRow = pivotRowForColumn(pivotColumn);

    if (pivotRow == -1) {

        qWarning() << "QSimplex: Unbounded problem!";

        return false;

    }

    // Normalize Pivot Row

    qreal pivot = valueAt(pivotRow, pivotColumn);

    if (pivot != 1.0)

        combineRows(pivotRow, pivotRow, (1.0 - pivot) / pivot);

    // Update other rows

    for (int row=0; row < rows; ++row) {

        if (row == pivotRow)

            continue;

        combineRows(row, pivotRow, -1 * valueAt(row, pivotColumn));

    }

    // Update first column

    setValueAt(pivotRow, 0, pivotColumn);

    //    dumpMatrix();

    //    qDebug("------------ end of iteration --------------\n");

    return true;

}

/*!

  \internal

  Both solveMin and solveMax are interfaces to this method.

  The enum solverFactor admits 2 values: Minimum (-1) and Maximum (+1).

  This method sets the original objective and runs the second phase

  Simplex to obtain the optimal solution for the problem. As the internal

  simplex solver is only able to _maximize_ objectives, we handle the

  minimization case by inverting the original objective and then

  maximizing it.

*/

qreal QSimplex::solver(solverFactor factor)

{

    // Remove old objective

    clearRow(0);

    // Set new objective in the first row of the simplex matrix

    qreal resultOffset = 0;