## Step 1: Define the symbolic representation of the problem
The problem involves minimizing an objective function subject to several constraints. The variables are:
- $x_0$: hours worked by Paul
- $x_1$: hours worked by John
- $x_2$: hours worked by George
- $x_3$: hours worked by Jean

The objective function to minimize is: $8x_0 + 5x_1 + 5x_2 + x_3$

## Step 2: List the constraints
The constraints given are:
- $11.95x_0 \leq 251$ (Implicitly, $x_0$ has no upper bound from this, but $11.95$ is a coefficient, not a constraint directly)
- $10.16x_0 \leq 115$
- $0.14x_0 \leq 81$
- $9.41x_1 \leq 251$
- $1.95x_1 \leq 115$
- $11.97x_1 \leq 81$
- $1.0x_2 \leq 251$
- $0.61x_2 \leq 115$
- $10.43x_2 \leq 81$
- $9.61x_3 \leq 251$
- $5.35x_3 \leq 115$
- $9.93x_3 \leq 81$
- $9.41x_1 + 9.61x_3 \geq 27$
- $1.0x_2 + 9.61x_3 \geq 21$
- $9.41x_1 + 1.0x_2 \geq 25$
- $11.95x_0 + 9.41x_1 \geq 47$
- $11.95x_0 + 1.0x_2 + 9.61x_3 \geq 49$
- $11.95x_0 + 9.41x_1 + 1.0x_2 + 9.61x_3 \geq 49$
- $10.16x_0 + 5.35x_3 \geq 24$
- $0.61x_2 + 5.35x_3 \geq 17$
- $1.95x_1 + 5.35x_3 \geq 9$
- $1.95x_1 + 0.61x_2 + 5.35x_3 \geq 28$
- $10.16x_0 + 1.95x_1 + 5.35x_3 \geq 28$
- $1.95x_1 + 0.61x_2 + 5.35x_3 \geq 28$
- $10.16x_0 + 1.95x_1 + 5.35x_3 \geq 28$
- $10.16x_0 + 1.95x_1 + 0.61x_2 + 5.35x_3 \geq 28$
- $10.43x_2 + 9.93x_3 \geq 19$
- $0.14x_0 + 10.43x_2 \geq 20$
- $11.97x_1 + 9.93x_3 \geq 8$
- $0.14x_0 + 9.93x_3 \geq 9$
- $0.14x_0 + 11.97x_1 + 10.43x_2 \geq 19$
- $0.14x_0 + 11.97x_1 + 10.43x_2 + 9.93x_3 \geq 19$
- $6x_0 - 8x_2 \geq 0$
- $8x_1 - 8x_3 \geq 0$
- $11.95x_0 + 9.41x_1 + 9.61x_3 \leq 162$
- $1.95x_1 + 0.61x_2 \leq 89$
- $1.95x_1 + 0.61x_2 + 5.35x_3 \leq 94$
- $10.16x_0 + 0.61x_2 + 5.35x_3 \leq 89$
- $10.16x_0 + 1.95x_1 + 0.61x_2 \leq 108$
- $11.97x_1 + 9.93x_3 \leq 32$
- $0.14x_0 + 10.43x_2 \leq 26$

## Step 3: Convert the problem into Gurobi code
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Paul
x1 = m.addVar(name="x1", lb=0)  # hours worked by John
x2 = m.addVar(name="x2", lb=0)  # hours worked by George
x3 = m.addVar(name="x3", lb=0)  # hours worked by Jean

# Define the objective function
m.setObjective(8 * x0 + 5 * x1 + 5 * x2 + x3, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(11.95 * x0 <= 251)
m.addConstr(10.16 * x0 <= 115)
m.addConstr(0.14 * x0 <= 81)

m.addConstr(9.41 * x1 <= 251)
m.addConstr(1.95 * x1 <= 115)
m.addConstr(11.97 * x1 <= 81)

m.addConstr(1.0 * x2 <= 251)
m.addConstr(0.61 * x2 <= 115)
m.addConstr(10.43 * x2 <= 81)

m.addConstr(9.61 * x3 <= 251)
m.addConstr(5.35 * x3 <= 115)
m.addConstr(9.93 * x3 <= 81)

m.addConstr(9.41 * x1 + 9.61 * x3 >= 27)
m.addConstr(1.0 * x2 + 9.61 * x3 >= 21)
m.addConstr(9.41 * x1 + 1.0 * x2 >= 25)
m.addConstr(11.95 * x0 + 9.41 * x1 >= 47)
m.addConstr(11.95 * x0 + 1.0 * x2 + 9.61 * x3 >= 49)
m.addConstr(11.95 * x0 + 9.41 * x1 + 1.0 * x2 + 9.61 * x3 >= 49)

m.addConstr(10.16 * x0 + 5.35 * x3 >= 24)
m.addConstr(0.61 * x2 + 5.35 * x3 >= 17)
m.addConstr(1.95 * x1 + 5.35 * x3 >= 9)
m.addConstr(1.95 * x1 + 0.61 * x2 + 5.35 * x3 >= 28)
m.addConstr(10.16 * x0 + 1.95 * x1 + 5.35 * x3 >= 28)
m.addConstr(1.95 * x1 + 0.61 * x2 + 5.35 * x3 >= 28)
m.addConstr(10.16 * x0 + 1.95 * x1 + 5.35 * x3 >= 28)
m.addConstr(10.16 * x0 + 1.95 * x1 + 0.61 * x2 + 5.35 * x3 >= 28)

m.addConstr(10.43 * x2 + 9.93 * x3 >= 19)
m.addConstr(0.14 * x0 + 10.43 * x2 >= 20)
m.addConstr(11.97 * x1 + 9.93 * x3 >= 8)
m.addConstr(0.14 * x0 + 9.93 * x3 >= 9)
m.addConstr(0.14 * x0 + 11.97 * x1 + 10.43 * x2 >= 19)
m.addConstr(0.14 * x0 + 11.97 * x1 + 10.43 * x2 + 9.93 * x3 >= 19)

m.addConstr(6 * x0 - 8 * x2 >= 0)
m.addConstr(8 * x1 - 8 * x3 >= 0)

m.addConstr(11.95 * x0 + 9.41 * x1 + 9.61 * x3 <= 162)
m.addConstr(1.95 * x1 + 0.61 * x2 <= 89)
m.addConstr(1.95 * x1 + 0.61 * x2 + 5.35 * x3 <= 94)
m.addConstr(10.16 * x0 + 0.61 * x2 + 5.35 * x3 <= 89)
m.addConstr(10.16 * x0 + 1.95 * x1 + 0.61 * x2 <= 108)
m.addConstr(11.97 * x1 + 9.93 * x3 <= 32)
m.addConstr(0.14 * x0 + 10.43 * x2 <= 26)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## Step 4: Symbolic Representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Paul'), 
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by George'), 
        ('x3', 'hours worked by Jean')
    ], 
    'objective_function': '8*x0 + 5*x1 + 5*x2 + x3', 
    'constraints': [
        '11.95*x0 <= 251',
        '10.16*x0 <= 115',
        '0.14*x0 <= 81',
        '9.41*x1 <= 251',
        '1.95*x1 <= 115',
        '11.97*x1 <= 81',
        '1.0*x2 <= 251',
        '0.61*x2 <= 115',
        '10.43*x2 <= 81',
        '9.61*x3 <= 251',
        '5.35*x3 <= 115',
        '9.93*x3 <= 81',
        '9.41*x1 + 9.61*x3 >= 27',
        '1.0*x2 + 9.61*x3 >= 21',
        '9.41*x1 + 1.0*x2 >= 25',
        '11.95*x0 + 9.41*x1 >= 47',
        '11.95*x0 + 1.0*x2 + 9.61*x3 >= 49',
        '11.95*x0 + 9.41*x1 + 1.0*x2 + 9.61*x3 >= 49',
        '10.16*x0 + 5.35*x3 >= 24',
        '0.61*x2 + 5.35*x3 >= 17',
        '1.95*x1 + 5.35*x3 >= 9',
        '1.95*x1 + 0.61*x2 + 5.35*x3 >= 28',
        '10.16*x0 + 1.95*x1 + 5.35*x3 >= 28',
        '1.95*x1 + 0.61*x2 + 5.35*x3 >= 28',
        '10.16*x0 + 1.95*x1 + 5.35*x3 >= 28',
        '10.16*x0 + 1.95*x1 + 0.61*x2 + 5.35*x3 >= 28',
        '10.43*x2 + 9.93*x3 >= 19',
        '0.14*x0 + 10.43*x2 >= 20',
        '11.97*x1 + 9.93*x3 >= 8',
        '0.14*x0 + 9.93*x3 >= 9',
        '0.14*x0 + 11.97*x1 + 10.43*x2 >= 19',
        '0.14*x0 + 11.97*x1 + 10.43*x2 + 9.93*x3 >= 19',
        '6*x0 - 8*x2 >= 0',
        '8*x1 - 8*x3 >= 0',
        '11.95*x0 + 9.41*x1 + 9.61*x3 <= 162',
        '1.95*x1 + 0.61*x2 <= 89',
        '1.95*x1 + 0.61*x2 + 5.35*x3 <= 94',
        '10.16*x0 + 0.61*x2 + 5.35*x3 <= 89',
        '10.16*x0 + 1.95*x1 + 0.61*x2 <= 108',
        '11.97*x1 + 9.93*x3 <= 32',
        '0.14*x0 + 10.43*x2 <= 26'
    ]
}
```