```json
{
  "sym_variables": [
    ("x0", "hours worked by Jean"),
    ("x1", "hours worked by John"),
    ("x2", "hours worked by Peggy"),
    ("x3", "hours worked by George"),
    ("x4", "hours worked by Mary")
  ],
  "objective_function": "5*x0 + 3*x1 + 3*x2 + 2*x3 + 6*x4",
  "constraints": [
    "16*x1 + 17*x2 >= 56",
    "22*x0 + 17*x2 >= 33",
    "17*x2 + 23*x4 >= 55",
    "22*x0 + 16*x1 + 17*x2 >= 51",
    "22*x0 + 16*x1 + 17*x2 + 2*x3 + 23*x4 >= 51",
    "-2*x0 + 8*x4 >= 0",
    "16*x1 + 2*x3 <= 254",
    "16*x1 + 17*x2 <= 96",
    "22*x0 + 16*x1 <= 170",
    "22*x0 + 2*x3 <= 94",
    "17*x2 + 23*x4 <= 87",
    "22*x0 + 23*x4 <= 202",
    "2*x3 + 23*x4 <= 62",
    "17*x2 + 2*x3 <= 244"
  ]
}
```

```python
import gurobipy as gp
from gurobipy import GRB

try:
    # Create a new model
    model = gp.Model("minimize_cost")

    # Create variables
    x = {}
    x[0] = model.addVar(vtype=GRB.INTEGER, name="Jean")
    x[1] = model.addVar(vtype=GRB.INTEGER, name="John")
    x[2] = model.addVar(vtype=GRB.INTEGER, name="Peggy")
    x[3] = model.addVar(vtype=GRB.CONTINUOUS, name="George")
    x[4] = model.addVar(vtype=GRB.CONTINUOUS, name="Mary")


    # Set objective function
    model.setObjective(5*x[0] + 3*x[1] + 3*x[2] + 2*x[3] + 6*x[4], GRB.MINIMIZE)

    # Add constraints
    model.addConstr(16*x[1] + 17*x[2] >= 56, "c1")
    model.addConstr(22*x[0] + 17*x[2] >= 33, "c2")
    model.addConstr(17*x[2] + 23*x[4] >= 55, "c3")
    model.addConstr(22*x[0] + 16*x[1] + 17*x[2] >= 51, "c4")
    model.addConstr(22*x[0] + 16*x[1] + 17*x[2] + 2*x[3] + 23*x[4] >= 51, "c5")
    model.addConstr(-2*x[0] + 8*x[4] >= 0, "c6")
    model.addConstr(16*x[1] + 2*x[3] <= 254, "c7")
    model.addConstr(16*x[1] + 17*x[2] <= 96, "c8")
    model.addConstr(22*x[0] + 16*x[1] <= 170, "c9")
    model.addConstr(22*x[0] + 2*x[3] <= 94, "c10")
    model.addConstr(17*x[2] + 23*x[4] <= 87, "c11")
    model.addConstr(22*x[0] + 23*x[4] <= 202, "c12")
    model.addConstr(2*x[3] + 23*x[4] <= 62, "c13")
    model.addConstr(17*x[2] + 2*x[3] <= 244, "c14")


    # Optimize model
    model.optimize()

    if model.status == GRB.OPTIMAL:
        for v in model.getVars():
            print('%s %g' % (v.varName, v.x))
        print('Obj: %g' % model.objVal)

    elif model.status == GRB.INFEASIBLE:
        print('Model is infeasible')
        model.computeIIS()
        model.write("model.ilp")


except gp.GurobiError as e:
    print('Error code ' + str(e.errno) + ': ' + str(e))

except AttributeError:
    print('Encountered an attribute error')
```