```json
{
  "sym_variables": [
    ("x0", "hours worked by Laura"),
    ("x1", "hours worked by George"),
    ("x2", "hours worked by Dale"),
    ("x3", "hours worked by Jean"),
    ("x4", "hours worked by Peggy")
  ],
  "objective_function": "6.78*x0 + 6.02*x1 + 2.49*x2 + 5.75*x3 + 5.93*x4",
  "constraints": [
    "4*x0 + 9*x4 >= 35",
    "4*x0 + 14*x2 >= 62",
    "14*x2 + 12*x3 >= 37",
    "4*x1 + 14*x2 >= 38",
    "4*x0 + 12*x3 >= 55",
    "14*x2 + 12*x3 + 9*x4 >= 53",
    "4*x1 + 14*x2 + 12*x3 >= 53",
    "4*x0 + 4*x1 + 9*x4 >= 53",
    "4*x0 + 4*x1 + 12*x3 >= 53",
    "4*x0 + 12*x3 + 9*x4 >= 53",
    "14*x2 + 12*x3 + 9*x4 >= 61",
    "4*x1 + 14*x2 + 12*x3 >= 61",
    "4*x0 + 4*x1 + 9*x4 >= 61",
    "4*x0 + 4*x1 + 12*x3 >= 61",
    "4*x0 + 12*x3 + 9*x4 >= 61",
    "14*x2 + 12*x3 + 9*x4 >= 64",
    "4*x1 + 14*x2 + 12*x3 >= 64",
    "4*x0 + 4*x1 + 9*x4 >= 64",
    "4*x0 + 4*x1 + 12*x3 >= 64",
    "4*x0 + 12*x3 + 9*x4 >= 64",
    "14*x2 + 12*x3 + 9*x4 >= 39",
    "4*x1 + 14*x2 + 12*x3 >= 39",
    "4*x0 + 4*x1 + 9*x4 >= 39",
    "4*x0 + 4*x1 + 12*x3 >= 39",
    "4*x0 + 12*x3 + 9*x4 >= 39",
    "14*x2 + 12*x3 + 9*x4 >= 70",
    "4*x1 + 14*x2 + 12*x3 >= 70",
    "4*x0 + 4*x1 + 9*x4 >= 70",
    "4*x0 + 4*x1 + 12*x3 >= 70",
    "4*x0 + 12*x3 + 9*x4 >= 70",
    "4*x0 + 4*x1 + 14*x2 + 12*x3 + 9*x4 >= 70",
    "7*x0 - 7*x3 >= 0",
    "5*x0 - 2*x2 >= 0",
    "12*x3 + 9*x4 <= 158"
  ]
}
```

```python
import gurobipy as gp

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

# Create variables
laura = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="laura")
george = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="george")
dale = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="dale")
jean = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="jean")
peggy = model.addVar(lb=0, vtype=gp.GRB.INTEGER, name="peggy")


# Set objective function
model.setObjective(6.78 * laura + 6.02 * george + 2.49 * dale + 5.75 * jean + 5.93 * peggy, gp.GRB.MINIMIZE)

# Add constraints
model.addConstr(4 * laura + 9 * peggy >= 35)
model.addConstr(4 * laura + 14 * dale >= 62)
model.addConstr(14 * dale + 12 * jean >= 37)
model.addConstr(4 * george + 14 * dale >= 38)
model.addConstr(4 * laura + 12 * jean >= 55)
model.addConstr(14 * dale + 12 * jean + 9 * peggy >= 53)
model.addConstr(4 * george + 14 * dale + 12 * jean >= 53)
model.addConstr(4 * laura + 4 * george + 9 * peggy >= 53)
model.addConstr(4 * laura + 4 * george + 12 * jean >= 53)
model.addConstr(4 * laura + 12 * jean + 9 * peggy >= 53)
model.addConstr(14 * dale + 12 * jean + 9 * peggy >= 61)
model.addConstr(4 * george + 14 * dale + 12 * jean >= 61)
model.addConstr(4 * laura + 4 * george + 9 * peggy >= 61)
model.addConstr(4 * laura + 4 * george + 12 * jean >= 61)
model.addConstr(4 * laura + 12 * jean + 9 * peggy >= 61)
model.addConstr(14 * dale + 12 * jean + 9 * peggy >= 64)
model.addConstr(4 * george + 14 * dale + 12 * jean >= 64)
model.addConstr(4 * laura + 4 * george + 9 * peggy >= 64)
model.addConstr(4 * laura + 4 * george + 12 * jean >= 64)
model.addConstr(4 * laura + 12 * jean + 9 * peggy >= 64)
model.addConstr(14 * dale + 12 * jean + 9 * peggy >= 39)
model.addConstr(4 * george + 14 * dale + 12 * jean >= 39)
model.addConstr(4 * laura + 4 * george + 9 * peggy >= 39)
model.addConstr(4 * laura + 4 * george + 12 * jean >= 39)
model.addConstr(4 * laura + 12 * jean + 9 * peggy >= 39)
model.addConstr(14 * dale + 12 * jean + 9 * peggy >= 70)
model.addConstr(4 * george + 14 * dale + 12 * jean >= 70)
model.addConstr(4 * laura + 4 * george + 9 * peggy >= 70)
model.addConstr(4 * laura + 4 * george + 12 * jean >= 70)
model.addConstr(4 * laura + 12 * jean + 9 * peggy >= 70)
model.addConstr(4 * laura + 4 * george + 14 * dale + 12 * jean + 9 * peggy >= 70)
model.addConstr(7 * laura - 7 * jean >= 0)
model.addConstr(5 * laura - 2 * dale >= 0)
model.addConstr(12 * jean + 9 * peggy <= 158)


# 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}")

```