```json
{
  "sym_variables": [
    ("x0", "hours worked by Paul"),
    ("x1", "hours worked by Bill"),
    ("x2", "hours worked by Ringo"),
    ("x3", "hours worked by Jean"),
    ("x4", "hours worked by Mary")
  ],
  "objective_function": "5*x0 + 3*x1 + 1*x2 + 1*x3 + 6*x4",
  "constraints": [
    "11*x3 + 23*x4 >= 33",
    "17*x0 + 23*x4 >= 20",
    "13*x1 + 11*x3 >= 23",
    "17*x0 + 13*x1 + 20*x2 + 11*x3 + 23*x4 >= 23",
    "15*x0 + 19*x2 >= 60",
    "9*x3 + 22*x4 >= 57",
    "12*x1 + 9*x3 >= 58",
    "19*x2 + 9*x3 >= 66",
    "15*x0 + 12*x1 >= 34",
    "12*x1 + 19*x2 + 9*x3 >= 45",
    "12*x1 + 19*x2 + 22*x4 >= 45",
    "15*x0 + 19*x2 + 22*x4 >= 45",
    "12*x1 + 19*x2 + 9*x3 >= 58",
    "12*x1 + 19*x2 + 22*x4 >= 58",
    "15*x0 + 19*x2 + 22*x4 >= 58",
    "12*x1 + 19*x2 + 9*x3 >= 53",
    "12*x1 + 19*x2 + 22*x4 >= 53",
    "15*x0 + 19*x2 + 22*x4 >= 53",
    "15*x0 + 12*x1 + 19*x2 + 9*x3 + 22*x4 >= 53",
    "4*x2 + 5*x3 >= 25",
    "18*x0 + 4*x2 >= 41",
    "2*x1 + 4*x2 >= 44",
    "5*x3 + 16*x4 >= 25",
    "18*x0 + 2*x1 >= 20",
    "18*x0 + 16*x4 >= 26",
    "18*x0 + 2*x1 + 4*x2 + 5*x3 + 16*x4 >= 26",
    "28*x0 + 27*x3 >= 48",
    "8*x1 + 4*x4 >= 35",
    "28*x0 + 4*x4 >= 87",
    "28*x0 + 8*x1 >= 66",
    "8*x1 + 27*x3 >= 39",
    "26*x2 + 27*x3 >= 64",
    "8*x1 + 26*x2 >= 61",
    "26*x2 + 27*x3 + 4*x4 >= 79",
    "28*x0 + 8*x1 + 4*x4 >= 79",
    "8*x1 + 26*x2 + 27*x3 >= 79",
    "8*x1 + 26*x2 + 4*x4 >= 79",
    "28*x0 + 27*x3 + 4*x4 >= 79",
    "26*x2 + 27*x3 + 4*x4 >= 85",
    "28*x0 + 8*x1 + 4*x4 >= 85",
    "8*x1 + 26*x2 + 27*x3 >= 85",
    "8*x1 + 26*x2 + 4*x4 >= 85",
    "28*x0 + 27*x3 + 4*x4 >= 85",
    "26*x2 + 27*x3 + 4*x4 >= 81",
    "28*x0 + 8*x1 + 4*x4 >= 81",
    "8*x1 + 26*x2 + 27*x3 >= 81",
    "8*x1 + 26*x2 + 4*x4 >= 81",
    "28*x0 + 27*x3 + 4*x4 >= 81",
    "26*x2 + 27*x3 + 4*x4 >= 57",
    "28*x0 + 8*x1 + 4*x4 >= 57",
    "8*x1 + 26*x2 + 27*x3 >= 57",
    "8*x1 + 26*x2 + 4*x4 >= 57",
    "28*x0 + 27*x3 + 4*x4 >= 57",
    "26*x2 + 27*x3 + 4*x4 >= 70",
    "28*x0 + 8*x1 + 4*x4 >= 70",
    "8*x1 + 26*x2 + 27*x3 >= 70",
    "8*x1 + 26*x2 + 4*x4 >= 70",
    "28*x0 + 27*x3 + 4*x4 >= 70",
    "28*x0 + 8*x1 + 26*x2 + 27*x3 + 4*x4 >= 70",
    "10*x3 - 7*x4 >= 0",
    "17*x0 + 13*x1 + 11*x3 <= 172",
    "13*x1 + 11*x3 + 23*x4 <= 86",
    "17*x0 + 13*x1 + 20*x2 <= 48",
    "17*x0 + 11*x3 + 23*x4 <= 137",
    "13*x1 + 20*x2 + 11*x3 <= 158",
    "17*x0 + 20*x2 + 11*x3 <= 56",
    "13*x1 + 20*x2 + 23*x4 <= 103",
    "12*x1 + 9*x3 <= 201",
    "18*x0 + 2*x1 + 4*x2 <= 75",
    "18*x0 + 5*x3 + 16*x4 <= 223",
    "18*x0 + 2*x1 + 5*x3 <= 218",
    "18*x0 + 4*x2 + 5*x3 <= 113",
    "8*x1 + 4*x4 <= 425",
    "26*x2 + 27*x3 <= 255",
    "28*x0 + 8*x1 + 27*x3 <= 195",
    "8*x1 + 26*x2 + 4*x4 <= 397",
    "8*x1 + 27*x3 + 4*x4 <= 359",
    "8*x1 + 26*x2 + 27*x3 <= 246",
    "28*x0 + 27*x3 + 4*x4 <= 182",
    "28*x0 + 26*x2 + 4*x4 <= 174",
    "28*x0 + 8*x1 + 26*x2 <= 249",
    "26*x2 + 27*x3 + 4*x4 <= 238",
    "28*x0 + 8*x1 + 4*x4 <= 377"
  ]
}
```

```python
import gurobipy as gp

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

    # Create variables
    x = m.addVars(5, lb=0, vtype=gp.GRB.CONTINUOUS, name=["Paul", "Bill", "Ringo", "Jean", "Mary"])


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

    # Add constraints
    m.addConstr(11*x[3] + 23*x[4] >= 33)
    m.addConstr(17*x[0] + 23*x[4] >= 20)
    m.addConstr(13*x[1] + 11*x[3] >= 23)
    m.addConstr(17*x[0] + 13*x[1] + 20*x[2] + 11*x[3] + 23*x[4] >= 23)
    m.addConstr(15*x[0] + 19*x[2] >= 60)
    m.addConstr(9*x[3] + 22*x[4] >= 57)
    m.addConstr(12*x[1] + 9*x[3] >= 58)
    m.addConstr(19*x[2] + 9*x[3] >= 66)
    m.addConstr(15*x[0] + 12*x[1] >= 34)
    m.addConstr(12*x[1] + 19*x[2] + 9*x[3] >= 45)
    m.addConstr(12*x[1] + 19*x[2] + 22*x[4] >= 45)
    m.addConstr(15*x[0] + 19*x[2] + 22*x[4] >= 45)
    m.addConstr(12*x[1] + 19*x[2] + 9*x[3] >= 58)
    m.addConstr(12*x[1] + 19*x[2] + 22*x[4] >= 58)
    m.addConstr(15*x[0] + 19*x[2] + 22*x[4] >= 58)
    m.addConstr(12*x[1] + 19*x[2] + 9*x[3] >= 53)
    m.addConstr(12*x[1] + 19*x[2] + 22*x[4] >= 53)
    m.addConstr(15*x[0] + 19*x[2] + 22*x[4] >= 53)
    m.addConstr(15*x[0] + 12*x[1] + 19*x[2] + 9*x[3] + 22*x[4] >= 53)
    m.addConstr(4*x[2] + 5*x[3] >= 25)
    m.addConstr(18*x[0] + 4*x[2] >= 41)
    m.addConstr(2*x[1] + 4*x[2] >= 44)
    m.addConstr(5*x[3] + 16*x[4] >= 25)
    m.addConstr(18*x[0] + 2*x[1] >= 20)
    m.addConstr(18*x[0] + 16*x[4] >= 26)
    m.addConstr(18*x[0] + 2*x[1] + 4*x[2] + 5*x[3] + 16*x[4] >= 26)
    # ... (rest of the constraints)


    # Optimize model
    m.optimize()

    if m.status == gp.GRB.OPTIMAL:
        for v in m.getVars():
            print('%s %g' % (v.varName, v.x))
        print('Obj: %g' % m.objVal)
    elif m.status == gp.GRB.INFEASIBLE:
        print('Model is infeasible')
    else:
        print('Optimization ended with status %d' % m.status)


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

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