```json
{
  "sym_variables": [
    ("x0", "hours worked by Laura"),
    ("x1", "hours worked by George"),
    ("x2", "hours worked by Bobby"),
    ("x3", "hours worked by Jean"),
    ("x4", "hours worked by Ringo")
  ],
  "objective_function": "1.78 * x0 + 3.84 * x1 + 5.67 * x2 + 7.82 * x3 + 4.85 * x4",
  "constraints": [
    "6 * x0 + 11 * x4 >= 34",
    "6 * x0 + 6 * x1 >= 26",
    "7 * x3 + 11 * x4 >= 36",
    "6 * x0 + 2 * x2 >= 24",
    "6 * x1 + 7 * x3 >= 20",
    "6 * x0 + 6 * x1 + 2 * x2 >= 37",
    "6 * x1 + 2 * x2 + 11 * x4 >= 37",
    "6 * x1 + 7 * x3 + 11 * x4 >= 37",
    "6 * x1 + 4 * x2 >= 14",
    "6 * x0 + 10 * x3 >= 27",
    "6 * x0 + 1 * x1 >= 16",
    "1 * x1 + 2 * x4 >= 27",
    "4 * x2 + 2 * x4 >= 22",
    "10 * x3 + 2 * x4 >= 34",
    "6 * x0 + 10 * x3 + 2 * x4 >= 35",  
    "4 * x2 + 10 * x3 + 2 * x4 >= 35",
    "6 * x0 + 4 * x2 + 2 * x4 >= 35",
    "1 * x1 + 4 * x2 + 2 * x4 >= 35",
    "6 * x0 + 1 * x1 + 2 * x4 >= 35",
    "6 * x0 + 1 * x1 + 10 * x3 >= 35",
    "6 * x0 + 6 * x1 + 7 * x3 <= 79",
    "6 * x1 + 2 * x2 + 7 * x3 <= 200",
    "6 * x0 + 2 * x2 + 11 * x4 <= 180",
    "2 * x2 + 7 * x3 + 11 * x4 <= 46",
    "6 * x1 + 7 * x3 + 11 * x4 <= 178",
    "6 * x1 + 2 * x2 + 11 * x4 <= 140",
    "6 * x0 + 6 * x1 + 11 * x4 <= 102",
    "6 * x0 + 7 * x3 + 11 * x4 <= 128",
    "6 * x0 + 6 * x1 + 2 * x2 + 7 * x3 + 11 * x4 >= 37",
    "9 * x0 - 8 * x4 >= 0",
    "-2 * x0 + 5 * x3 >= 0",
    "x0, x1, x2, x3, x4 are integers"


  ]
}
```

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Minimize_Work_Hours")

# Create variables
laura = model.addVar(vtype=gp.GRB.INTEGER, name="laura")
george = model.addVar(vtype=gp.GRB.INTEGER, name="george")
bobby = model.addVar(vtype=gp.GRB.INTEGER, name="bobby")
jean = model.addVar(vtype=gp.GRB.INTEGER, name="jean")
ringo = model.addVar(vtype=gp.GRB.INTEGER, name="ringo")


# Set objective function
model.setObjective(1.78 * laura + 3.84 * george + 5.67 * bobby + 7.82 * jean + 4.85 * ringo, gp.GRB.MINIMIZE)

# Add constraints
model.addConstr(6 * laura + 11 * ringo >= 34)
model.addConstr(6 * laura + 6 * george >= 26)
model.addConstr(7 * jean + 11 * ringo >= 36)
model.addConstr(6 * laura + 2 * bobby >= 24)
model.addConstr(6 * george + 7 * jean >= 20)
model.addConstr(6 * laura + 6 * george + 2 * bobby >= 37)
model.addConstr(6 * george + 2 * bobby + 11 * ringo >= 37)
model.addConstr(6 * george + 7 * jean + 11 * ringo >= 37)
model.addConstr(6 * george + 4 * bobby >= 14)
model.addConstr(6 * laura + 10 * jean >= 27)
model.addConstr(6 * laura + 1 * george >= 16)
model.addConstr(1 * george + 2 * ringo >= 27)
model.addConstr(4 * bobby + 2 * ringo >= 22)
model.addConstr(10 * jean + 2 * ringo >= 34)
model.addConstr(6 * laura + 10 * jean + 2 * ringo >= 35)
model.addConstr(4 * bobby + 10 * jean + 2 * ringo >= 35)
model.addConstr(6 * laura + 4 * bobby + 2 * ringo >= 35)
model.addConstr(1 * george + 4 * bobby + 2 * ringo >= 35)
model.addConstr(6 * laura + 1 * george + 2 * ringo >= 35)
model.addConstr(6 * laura + 1 * george + 10 * jean >= 35)
model.addConstr(6 * laura + 6 * george + 7 * jean <= 79)
model.addConstr(6 * george + 2 * bobby + 7 * jean <= 200)
model.addConstr(6 * laura + 2 * bobby + 11 * ringo <= 180)
model.addConstr(2 * bobby + 7 * jean + 11 * ringo <= 46)
model.addConstr(6 * george + 7 * jean + 11 * ringo <= 178)
model.addConstr(6 * george + 2 * bobby + 11 * ringo <= 140)
model.addConstr(6 * laura + 6 * george + 11 * ringo <= 102)
model.addConstr(6 * laura + 7 * jean + 11 * ringo <= 128)
model.addConstr(6 * laura + 6 * george + 2 * bobby + 7 * jean + 11 * ringo >= 37)
model.addConstr(9 * laura - 8 * ringo >= 0)
model.addConstr(-2 * laura + 5 * jean >= 0)



# Optimize model
model.optimize()

# Print results
if model.status == gp.GRB.OPTIMAL:
    print('Obj: %g' % model.objVal)
    for v in model.getVars():
        print('%s %g' % (v.varName, v.x))
elif model.status == gp.GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    print(f"Optimization ended with status {model.status}")

```
