/// <summary>
/// If there was a first phase and the optimum of the objective function was 0 the model should be transformed to such a dictionary form on which the second phase can be executed.
/// This function does the necessary changes on the model.
/// </summary>
/// <param name="model">The LP model wanted to be transformed to be eligible to the execution of the second phase.</param>
/// <returns>The transformed LP model.</returns>
private static LPModel ToSecondPhaseDictionaryForm(this LPModel model)
{
#region Throw out the 'var0 = 0' shaped constraint if any
var constraintToRemove = model.Constraints.Where(constraint => constraint.LeftSide.Count == 1 && constraint.LeftSide.Any(term => term.Variable.Value.Index == 0) &&
(constraint.RightSide.Count == 1 || constraint.RightSide.Any(term => term.Constant && term.SignedCoefficient == 0)) || constraint.RightSide.Count == 0)
.SingleOrDefault();
if (constraintToRemove != null)
{
model.Constraints.Remove(constraintToRemove);
}
#endregion
#region If var0 is a basis variable it will be exchanged with a non-basis variable by a pivot step
var constraintWithVar0Basis = model.Constraints.Where(constraint => constraint.LeftSide.Count == 1 && constraint.LeftSide.Any(term => term.Variable.Value.Index == 0))
.SingleOrDefault();
var var0 = new Variable {
Name = model.AllVariables.First().Name, Index = 0
};
if (constraintWithVar0Basis != null)
{
// choosing the variable has a negative coefficient and the smallest index
var newBasisVariable = constraintWithVar0Basis.RightSide
.Where(term => term.Variable.Value.Index == constraintWithVar0Basis.RightSide
.Where(t => t.Variable.HasValue /*&& t.SignedCoefficient < 0*/)
.Min(t => t.Variable.Value.Index))
.Single().Variable.Value;
model.MakePivotStep(newBasisVariable, var0, withoutObjectiveFunction: true);
}
#endregion
#region Throw out the rest of the var0 occurences
model.Constraints.ForAll(constraint =>
{
var foundVar0 = constraint.RightSide.Where(term => term.Variable?.Equals(new Variable {
Name = model.AllVariables.First().Name, Index = 0
}) ?? false).SingleOrDefault();
if (foundVar0 != null)
{
constraint.RightSide.Remove(foundVar0);
}
});
model.InterpretationRanges.Remove(model.InterpretationRanges.Single(range => range.LeftSide.Single().Variable?.Equals(var0) ?? false));
#endregion
#region Set back the original objective function exhanging the basis variables with their equivaltent expressions from the dictionary (right sides)
model.Objective = model.TmpObjective;
model.Constraints.ForAll(constraint =>
{
var basisVariable = constraint.LeftSide.Single(term => term.Variable.HasValue).Variable.Value;
model.Objective.Function.ReplaceVarWithExpression(basisVariable, constraint.RightSide);
});
#endregion
return(model);
}
/// <summary>
/// Runs the simplex algoritm on the given LP model.
/// </summary>
/// <param name="model">The LP model on which the simplex algoritm will be executed.</param>
/// <returns></returns>
/// <exception cref="SimplexAlgorithmExectionException"></exception>
public static LPModel RunSimplex(this LPModel model)
{
while (!model.AllObjectiveFunctionVariableHasNegativeCoefficient())
{
#region Choosing new basis variable by Bland - let1s name its index as k
var kIndex = model.Objective.Function.RightSide.Where(term => term.Variable.HasValue && term.SignedCoefficient > 0).Min(term => term.Variable.Value.Index);
var stepInVariable = model.Objective.Function.RightSide.Where(term => term.Variable.HasValue && term.Variable.Value.Index == kIndex).Single().Variable.Value;
#endregion
#region All k-indexed variable has positive coefficient in the constraints? YES - stop, no limit, NO - continue
Func <Term, bool> hasKIndex = term => term.Variable?.Equals(stepInVariable) ?? false;
IEnumerable <Term> termsWithKIndexedVariables = model.Constraints.Where(constraint => constraint.RightSide.Any(term => hasKIndex(term)))
.Select(constraint => constraint.RightSide.Where(term => hasKIndex(term)).Single());
bool allKIndexedHasPositiveIndex = termsWithKIndexedVariables.All(term => term.SignedCoefficient > 0);
if (allKIndexedHasPositiveIndex)
{
throw new SimplexAlgorithmExectionException(SimplexAlgorithmExectionErrorType.NotLimited);
}
#endregion
#region Determining the variable which will leave the base by finding the smallest quotient
Equation selectedConstraint = null;
Rational?smallestQuotient = null;
model.Constraints.Where(constraint => constraint.RightSide.Any(term => hasKIndex(term) && term.SignedCoefficient < 0))
.ForAll(constraint =>
{
// This value must be non-negative - if not the dictionary were not valid
var constraintsConstantValue = constraint.RightSide.Where(term => term.Constant).SingleOrDefault()?.SignedCoefficient ?? 0;
var kIndexedVariablesCoefficient = constraint.RightSide.Where(term => hasKIndex(term)).Single().SignedCoefficient;
// If this is the fist ratio, we will store it automatically (smallestQuotient doesn't have a value yet
var smallestQuotientFound = !smallestQuotient.HasValue || ((constraintsConstantValue / kIndexedVariablesCoefficient.Abs()) < smallestQuotient);
if (smallestQuotientFound)
{
selectedConstraint = constraint;
smallestQuotient = constraintsConstantValue / kIndexedVariablesCoefficient.Abs();
}
});
var stepOutVariable = selectedConstraint.LeftSide.Single().Variable.Value;
#endregion
#region Making a pivot step
model.MakePivotStep(stepInVariable, stepOutVariable);
#endregion
}
return(model);
}
/// <summary>
/// Runs the dual simplex algoritm on the given LP model.
/// </summary>
/// <param name="lpModel">The LP model on wich the dual simplex method will be executed.</param>
/// <returns>The LP model</returns>
public static LPModel DualSimplex(this LPModel model)
{
while (!model.AllBasisVariableHaveNonNegativeValuesInTheDictionary())
{
#region Choosing variable will step out from the base - choosing the dictionary row having the most negative index - we will choose the lowest index anyway
var stepOutVariable = model.Constraints.Where(dictionaryRow => (dictionaryRow.RightSide.SingleOrDefault(term => term.Constant)?.SignedCoefficient ?? 0) < 0)
.Select(dictionaryRow => new { Variable = dictionaryRow.LeftSide.Single().Variable.Value, Value = dictionaryRow.RightSide.Single(term => term.Constant).SignedCoefficient })
.OrderBy(variableValuePair => variableValuePair.Value)
.ThenBy(variableValuePair => variableValuePair.Variable.Index)
.First().Variable;
#endregion
var rowWithStepOutVariable = model.Constraints.Single(row => row.LeftSide.Single().Variable.Value.Equals(stepOutVariable));
if (rowWithStepOutVariable.RightSide.Where(term => !term.Constant).All(term => term.SignedCoefficient < 0))
{
throw new SimplexAlgorithmExectionException(SimplexAlgorithmExectionErrorType.NoSolution);
}
#region Choosing the variable will step in the basis - by finding the smallest quotient
var termsWithPositiveCoefficient = rowWithStepOutVariable.RightSide.Where(term => term.SignedCoefficient > 0 && !term.Constant);
var varsWithQuotient = new Dictionary <Variable, Rational>();
termsWithPositiveCoefficient.ForAll(term =>
{
var functionTermsCoefficient = model.Objective.Function.RightSide.SingleOrDefault(functionTerm => !functionTerm.Constant && functionTerm.Variable.Value.Equals(term.Variable.Value))?.SignedCoefficient ?? 0;
varsWithQuotient.Add(term.Variable.Value, functionTermsCoefficient / term.SignedCoefficient);
});
var stepInVariable = varsWithQuotient.OrderBy(kv => kv.Value).ThenBy(kv => kv.Key.Index).First().Key;
#endregion
#region Making a pivot step
model.MakePivotStep(stepInVariable, stepOutVariable);
#endregion
}
return(model);
}