diff options
Diffstat (limited to 'src/gui/graphicsview/qsimplex_p.cpp')
| -rw-r--r-- | src/gui/graphicsview/qsimplex_p.cpp | 173 |
1 files changed, 156 insertions, 17 deletions
diff --git a/src/gui/graphicsview/qsimplex_p.cpp b/src/gui/graphicsview/qsimplex_p.cpp index e3a991e..1ba24a3 100644 --- a/src/gui/graphicsview/qsimplex_p.cpp +++ b/src/gui/graphicsview/qsimplex_p.cpp @@ -48,6 +48,32 @@ QT_BEGIN_NAMESPACE +/*! + \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. +*/ + QSimplex::QSimplex() : objective(0), rows(0), columns(0), firstArtificial(0), matrix(0) { } @@ -84,15 +110,31 @@ void QSimplex::clearDataStructures() objective = 0; } +/*! + 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 constraints = newConstraints; - // Set Variables direct mapping + /////////////////////////////////////// + // 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 += \ @@ -100,12 +142,25 @@ bool QSimplex::setConstraints(const QList<QSimplexConstraint *> newConstraints) 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; @@ -138,12 +193,18 @@ bool QSimplex::setConstraints(const QList<QSimplexConstraint *> newConstraints) } } + // 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(); - // Matrix + ///////////////////////////// + // Fill the Simplex matrix // + ///////////////////////////// // One for each variable plus the Basic and BFS columns (first and last) columns = variableIndex + 2; @@ -188,24 +249,61 @@ bool QSimplex::setConstraints(const QList<QSimplexConstraint *> newConstraints) setValueAt(i, columns - 1, c->constant); } - // Set temporary objective: -1 * sum_of_artificial_vars + // 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) { qWarning() << "QSimplex: No feasible solution!"; clearDataStructures(); return false; } - // Remove artificial variables + // Remove artificial variables. We already have a feasible + // solution for the first problem, thus we don't need them + // anymore. clearColumns(firstArtificial, columns - 2); + + #ifdef QT_DEBUG + // Ensure that at the end of the simplex each row should either: + // - Have a positive value on the column associated to its variable, or + // - Have zero values in all columns. + // + // This avoids a regression where restrictions would be lost + // due to randomness in the pivotRowForColumn method. + for (int i = 1; i < rows; ++i) { + int variableIndex = valueAt(i, 0); + if (valueAt(i, variableIndex) > 0) + continue; + + for (int j = 1; j < columns; ++j) { + Q_ASSERT(valueAt(i, j) == 0); + } + } + #endif + 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(); @@ -235,23 +333,21 @@ void QSimplex::clearColumns(int first, int last) void QSimplex::dumpMatrix() { - printf("---- Simplex Matrix ----\n"); + qDebug("---- Simplex Matrix ----\n"); - printf(" "); + QString str(QLatin1String(" ")); for (int j = 0; j < columns; ++j) - printf(" <% 2d >", j); - printf("\n"); - + str += QString::fromAscii(" <%1 >").arg(j, 2); + qDebug("%s", qPrintable(str)); for (int i = 0; i < rows; ++i) { - printf("Row %2d:", i); + str = QString::fromAscii("Row %1:").arg(i, 2); qreal *row = matrix + i * columns; - for (int j = 0; j < columns; ++j) { - printf(" % 2.2f", row[j]); - } - printf("\n"); + for (int j = 0; j < columns; ++j) + str += QString::fromAscii("%1").arg(row[j], 7, 'f', 2); + qDebug("%s", qPrintable(str)); } - printf("------------------------\n\n"); + qDebug("------------------------\n"); } void QSimplex::combineRows(int toIndex, int fromIndex, qreal factor) @@ -292,6 +388,23 @@ int QSimplex::findPivotColumn() 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); // ### @@ -306,6 +419,8 @@ int QSimplex::pivotRowForColumn(int column) if (quotient < min) { min = quotient; minIndex = i; + } else if ((quotient == min) && (valueAt(i, 0) > valueAt(minIndex, 0))) { + minIndex = i; } } @@ -320,6 +435,12 @@ void QSimplex::reducedRowEchelon() } } +/*! + \internal + + Does one iteration towards a better solution for the problem. + See 'solveMaxHelper'. +*/ bool QSimplex::iterate() { // Find Pivot column @@ -351,7 +472,7 @@ bool QSimplex::iterate() setValueAt(pivotRow, 0, pivotColumn); // dumpMatrix(); - // printf("------------ end of iteration --------------\n"); + // qDebug("------------ end of iteration --------------\n"); return true; } @@ -361,7 +482,13 @@ bool QSimplex::iterate() 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 @@ -381,16 +508,28 @@ qreal QSimplex::solver(solverFactor factor) return factor * valueAt(0, columns - 1); } +/*! + Minimize the original objective. +*/ qreal QSimplex::solveMin() { return solver(Minimum); } +/*! + Maximize the original objective. +*/ qreal QSimplex::solveMax() { return solver(Maximum); } +/*! + \internal + + Reads results from the simplified matrix and saves them in the + "result" member of each QSimplexVariable. +*/ void QSimplex::collectResults() { // All variables are zero unless overridden below. |