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### workforce4_c++.cpp

```/* Copyright 2019, Gurobi Optimization, LLC */

/* Assign workers to shifts; each worker may or may not be available on a
* particular day. We use Pareto optimization to solve the model:
* first, we minimize the linear sum of the slacks. Then, we constrain
* the sum of the slacks, and we minimize a quadratic objective that
* tries to balance the workload among the workers. */

#include "gurobi_c++.h"
#include <sstream>
using namespace std;

int solveAndPrint(GRBModel& model, GRBVar& totSlack,
int nWorkers, string* Workers,
GRBVar* totShifts) throw(GRBException);

int
main(int   argc,
char *argv[])
{
GRBEnv* env = 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];
int s, w;

for (w = 0; w < nWorkers; ++w) {
xCt++;

for (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
for (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
for (w = 0; w < nWorkers; ++w) {
ostringstream vname;

vname << Workers[w] << "TotShifts";
totShifts[w].set(GRB_StringAttr_VarName, vname.str());
}

GRBLinExpr lhs;

// Constraint: assign exactly shiftRequirements[s] workers
// to each shift s
for (s = 0; s < nShifts; ++s) {
lhs = 0;
lhs += slacks[s];

for (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 (s = 0; s < nShifts; ++s)
{
lhs += slacks[s];
}
model.addConstr(lhs == totSlack, "totSlack");

// Constraint: compute the total number of shifts for each worker
for (w = 0; w < nWorkers; ++w) {
lhs = 0;
for (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.
for (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);
}

// Constraint: compute the average number of shifts worked
lhs = 0;
for (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 (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
for (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;
}

for (int i = 0; i < xCt; ++i) {
delete[] x[i];
}
delete[] x;
delete[] slacks;
delete[] totShifts;
delete[] diffShifts;
delete env;

return 0;
}

int solveAndPrint(GRBModel& model,
GRBVar&   totSlack,
int       nWorkers,
string*   Workers,
GRBVar*   totShifts) throw(GRBException)
{
model.optimize();
int status = model.get(GRB_IntAttr_Status);

if ((status == GRB_INF_OR_UNBD) ||
(status == GRB_INFEASIBLE)  ||
(status == GRB_UNBOUNDED)     ) {
cout << "The model cannot be solved " <<
"because it is infeasible or unbounded" << endl;
return status;
}
if (status != GRB_OPTIMAL) {
cout << "Optimization was stopped with status " << status << endl;
return status;
}

// Print total slack and the number of shifts worked for each worker
cout << endl << "Total slack required: " <<
totSlack.get(GRB_DoubleAttr_X) << endl;
for (int w = 0; w < nWorkers; ++w) {
cout << Workers[w] << " worked " <<
totShifts[w].get(GRB_DoubleAttr_X) << " shifts" << endl;
}
cout << endl;

return status;
}
```