int
main(int argc,
char *argv[])
{
GRBEnv* env = 0;
GRBConstr* c = 0;
GRBVar* v = 0;
GRBVar** x = 0;
GRBVar* slacks = 0;
GRBVar* totShifts = 0;
GRBVar* diffShifts = 0;
int xCt = 0;
try
{
// Sample data
const int nShifts = 14;
const int nWorkers = 7;
// Sets of days and workers
string Shifts[] =
{ "Mon1", "Tue2", "Wed3", "Thu4", "Fri5", "Sat6",
"Sun7", "Mon8", "Tue9", "Wed10", "Thu11", "Fri12", "Sat13",
"Sun14" };
string Workers[] =
{ "Amy", "Bob", "Cathy", "Dan", "Ed", "Fred", "Gu" };
// Number of workers required for each shift
double shiftRequirements[] =
{ 3, 2, 4, 4, 5, 6, 5, 2, 2, 3, 4, 6, 7, 5 };
// Worker availability: 0 if the worker is unavailable for a shift
double availability[][nShifts] =
{ { 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1 },
{ 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0 },
{ 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 },
{ 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1 },
{ 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1 },
{ 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1 },
{ 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 } };
// Model
env = new GRBEnv();
GRBModel model = GRBModel(*env);
model.set(GRB_StringAttr_ModelName, "assignment");
// Assignment variables: x[w][s] == 1 if worker w is assigned
// to shift s. This is no longer a pure assignment model, so we must
// use binary variables.
x = new GRBVar*[nWorkers];
for (int w = 0; w < nWorkers; ++w)
{
x[w] = model.addVars(nShifts);
xCt++;
model.update();
for (int s = 0; s < nShifts; ++s)
{
ostringstream vname;
vname << Workers[w] << "." << Shifts[s];
x[w][s].set(GRB_DoubleAttr_UB, availability[w][s]);
x[w][s].set(GRB_CharAttr_VType, GRB_BINARY);
x[w][s].set(GRB_StringAttr_VarName, vname.str());
}
}
// Slack variables for each shift constraint so that the shifts can
// be satisfied
slacks = model.addVars(nShifts);
model.update();
for (int s = 0; s < nShifts; ++s)
{
ostringstream vname;
vname << Shifts[s] << "Slack";
slacks[s].set(GRB_StringAttr_VarName, vname.str());
}
// Variable to represent the total slack
GRBVar totSlack = model.addVar(0, GRB_INFINITY, 0, GRB_CONTINUOUS,
"totSlack");
// Variables to count the total shifts worked by each worker
totShifts = model.addVars(nWorkers);
model.update();
for (int w = 0; w < nWorkers; ++w)
{
ostringstream vname;
vname << Workers[w] << "TotShifts";
totShifts[w].set(GRB_StringAttr_VarName, vname.str());
}
// Update model to integrate new variables
model.update();
GRBLinExpr lhs;
// Constraint: assign exactly shiftRequirements[s] workers
// to each shift s
for (int s = 0; s < nShifts; ++s)
{
lhs = 0;
lhs += slacks[s];
for (int w = 0; w < nWorkers; ++w)
{
lhs += x[w][s];
}
model.addConstr(lhs == shiftRequirements[s], Shifts[s]);
}
// Constraint: set totSlack equal to the total slack
lhs = 0;
for (int s = 0; s < nShifts; ++s)
{
lhs += slacks[s];
}
model.addConstr(lhs == totSlack, "totSlack");
// Constraint: compute the total number of shifts for each worker
for (int w = 0; w < nWorkers; ++w) {
lhs = 0;
for (int s = 0; s < nShifts; ++s) {
lhs += x[w][s];
}
ostringstream vname;
vname << "totShifts" << Workers[w];
model.addConstr(lhs == totShifts[w], vname.str());
}
// Objective: minimize the total slack
GRBLinExpr obj = 0;
obj += totSlack;
model.setObjective(obj);
// Optimize
int status = solveAndPrint(model, totSlack, nWorkers, Workers, totShifts);
if (status != GRB_OPTIMAL)
{
return 1;
}
// Constrain the slack by setting its upper and lower bounds
totSlack.set(GRB_DoubleAttr_UB, totSlack.get(GRB_DoubleAttr_X));
totSlack.set(GRB_DoubleAttr_LB, totSlack.get(GRB_DoubleAttr_X));
// Variable to count the average number of shifts worked
GRBVar avgShifts =
model.addVar(0, GRB_INFINITY, 0, GRB_CONTINUOUS, "avgShifts");
// Variables to count the difference from average for each worker;
// note that these variables can take negative values.
diffShifts = model.addVars(nWorkers);
model.update();
for (int w = 0; w < nWorkers; ++w) {
ostringstream vname;
vname << Workers[w] << "Diff";
diffShifts[w].set(GRB_StringAttr_VarName, vname.str());
diffShifts[w].set(GRB_DoubleAttr_LB, -GRB_INFINITY);
}
// Update model to integrate new variables
model.update();
// Constraint: compute the average number of shifts worked
lhs = 0;
for (int w = 0; w < nWorkers; ++w) {
lhs += totShifts[w];
}
model.addConstr(lhs == nWorkers * avgShifts, "avgShifts");
// Constraint: compute the difference from the average number of shifts
for (int w = 0; w < nWorkers; ++w) {
lhs = 0;
lhs += totShifts[w];
lhs -= avgShifts;
ostringstream vname;
vname << Workers[w] << "Diff";
model.addConstr(lhs == diffShifts[w], vname.str());
}
// Objective: minimize the sum of the square of the difference from the
// average number of shifts worked
GRBQuadExpr qobj;
for (int w = 0; w < nWorkers; ++w) {
qobj += diffShifts[w] * diffShifts[w];
}
model.setObjective(qobj);
// Optimize
status = solveAndPrint(model, totSlack, nWorkers, Workers, totShifts);
if (status != GRB_OPTIMAL)
{
return 1;
}
}
catch (GRBException e)
{
cout << "Error code = " << e.getErrorCode() << endl;
cout << e.getMessage() << endl;
}
catch (...)
{
cout << "Exception during optimization" << endl;
}
delete[] c;
delete[] v;
for (int i = 0; i < xCt; ++i) {
delete[] x[i];
}
delete[] x;
delete[] slacks;
delete[] totShifts;
delete[] diffShifts;
delete env;
return 0;
}
