## Step 1: Define the symbolic representation of the problem
The problem involves minimizing an objective function subject to several constraints. The variables are 'hours worked by Dale', 'hours worked by Paul', 'hours worked by Laura', and 'hours worked by John', which we can denote as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Identify the objective function
The objective function to minimize is $9.78x_0 + 3.35x_1 + 4.5x_2 + 4.6x_3$.

## 3: List the constraints
The constraints are:
- $9x_0 \leq 262$
- $9x_1 \leq 262$
- $12x_2 \leq 262$
- $9x_3 \leq 262$
- $9x_0 \leq 320$
- $11x_1 \leq 320$
- $10x_2 \leq 320$
- $11x_3 \leq 320$
- $11x_0 \leq 103$
- $6x_1 \leq 103$
- $2x_2 \leq 103$
- $11x_3 \leq 103$
- $9x_0 + 12x_2 \geq 65$
- $9x_0 + 9x_3 \geq 40$
- $9x_0 + 9x_1 \geq 44$
- $9x_1 + 9x_3 \geq 65$
- $9x_1 + 12x_2 \geq 34$
- $9x_0 + 9x_1 + 12x_2 \geq 55$
- $9x_0 + 12x_2 + 9x_3 \geq 55$
- $9x_1 + 12x_2 + 9x_3 \geq 55$
- $9x_0 + 9x_1 + 12x_2 \geq 63$
- $9x_0 + 12x_2 + 9x_3 \geq 63$
- $9x_1 + 12x_2 + 9x_3 \geq 63$
- $9x_0 + 9x_1 + 12x_2 \geq 44$
- $9x_0 + 12x_2 + 9x_3 \geq 44$
- $9x_1 + 12x_2 + 9x_3 \geq 44$
- $9x_0 + 9x_1 + 12x_2 + 9x_3 \geq 44$
- $9x_0 + 10x_2 \geq 68$
- $11x_1 + 11x_3 \geq 66$
- $9x_0 + 11x_1 \geq 76$
- $9x_0 + 10x_2 + 11x_3 \geq 76$
- $9x_0 + 11x_1 + 10x_2 + 11x_3 \geq 76$
- $6x_1 + 2x_2 \geq 13$
- $2x_2 + 11x_3 \geq 13$
- $11x_0 + 2x_2 \geq 18$
- $11x_0 + 6x_1 + 2x_2 + 11x_3 \geq 18$
- $-10x_2 + 3x_3 \geq 0$
- $9x_1 + 9x_3 \leq 169$
- $9x_0 + 12x_2 \leq 178$
- $9x_0 + 9x_1 \leq 228$
- $9x_1 + 12x_2 + 9x_3 \leq 91$
- $9x_0 + 11x_1 + 11x_3 \leq 86$
- $6x_1 + 2x_2 + 11x_3 \leq 26$
- $11x_0 + 2x_2 + 11x_3 \leq 62$

## 4: Provide the symbolic representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Dale'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Laura'),
        ('x3', 'hours worked by John')
    ],
    'objective_function': '9.78*x0 + 3.35*x1 + 4.5*x2 + 4.6*x3',
    'constraints': [
        '9*x0 <= 262',
        '9*x1 <= 262',
        '12*x2 <= 262',
        '9*x3 <= 262',
        '9*x0 <= 320',
        '11*x1 <= 320',
        '10*x2 <= 320',
        '11*x3 <= 320',
        '11*x0 <= 103',
        '6*x1 <= 103',
        '2*x2 <= 103',
        '11*x3 <= 103',
        '9*x0 + 12*x2 >= 65',
        '9*x0 + 9*x3 >= 40',
        '9*x0 + 9*x1 >= 44',
        '9*x1 + 9*x3 >= 65',
        '9*x1 + 12*x2 >= 34',
        '9*x0 + 9*x1 + 12*x2 >= 55',
        '9*x0 + 12*x2 + 9*x3 >= 55',
        '9*x1 + 12*x2 + 9*x3 >= 55',
        '9*x0 + 9*x1 + 12*x2 >= 63',
        '9*x0 + 12*x2 + 9*x3 >= 63',
        '9*x1 + 12*x2 + 9*x3 >= 63',
        '9*x0 + 9*x1 + 12*x2 >= 44',
        '9*x0 + 12*x2 + 9*x3 >= 44',
        '9*x1 + 12*x2 + 9*x3 >= 44',
        '9*x0 + 9*x1 + 12*x2 + 9*x3 >= 44',
        '9*x0 + 10*x2 >= 68',
        '11*x1 + 11*x3 >= 66',
        '9*x0 + 11*x1 >= 76',
        '9*x0 + 10*x2 + 11*x3 >= 76',
        '9*x0 + 11*x1 + 10*x2 + 11*x3 >= 76',
        '6*x1 + 2*x2 >= 13',
        '2*x2 + 11*x3 >= 13',
        '11*x0 + 2*x2 >= 18',
        '11*x0 + 6*x1 + 2*x2 + 11*x3 >= 18',
        '-10*x2 + 3*x3 >= 0',
        '9*x1 + 9*x3 <= 169',
        '9*x0 + 12*x2 <= 178',
        '9*x0 + 9*x1 <= 228',
        '9*x1 + 12*x2 + 9*x3 <= 91',
        '9*x0 + 11*x1 + 11*x3 <= 86',
        '6*x1 + 2*x2 + 11*x3 <= 26',
        '11*x0 + 2*x2 + 11*x3 <= 62'
    ]
}
```

## 5: Implement the problem in Gurobi
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Dale
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Paul
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Laura
    x3 = model.addVar(name="x3", lb=0)  # hours worked by John

    # Objective function
    model.setObjective(9.78 * x0 + 3.35 * x1 + 4.5 * x2 + 4.6 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9 * x0 <= 262)
    model.addConstr(9 * x1 <= 262)
    model.addConstr(12 * x2 <= 262)
    model.addConstr(9 * x3 <= 262)
    model.addConstr(9 * x0 <= 320)
    model.addConstr(11 * x1 <= 320)
    model.addConstr(10 * x2 <= 320)
    model.addConstr(11 * x3 <= 320)
    model.addConstr(11 * x0 <= 103)
    model.addConstr(6 * x1 <= 103)
    model.addConstr(2 * x2 <= 103)
    model.addConstr(11 * x3 <= 103)
    model.addConstr(9 * x0 + 12 * x2 >= 65)
    model.addConstr(9 * x0 + 9 * x3 >= 40)
    model.addConstr(9 * x0 + 9 * x1 >= 44)
    model.addConstr(9 * x1 + 9 * x3 >= 65)
    model.addConstr(9 * x1 + 12 * x2 >= 34)
    model.addConstr(9 * x0 + 9 * x1 + 12 * x2 >= 55)
    model.addConstr(9 * x0 + 12 * x2 + 9 * x3 >= 55)
    model.addConstr(9 * x1 + 12 * x2 + 9 * x3 >= 55)
    model.addConstr(9 * x0 + 9 * x1 + 12 * x2 >= 63)
    model.addConstr(9 * x0 + 12 * x2 + 9 * x3 >= 63)
    model.addConstr(9 * x1 + 12 * x2 + 9 * x3 >= 63)
    model.addConstr(9 * x0 + 9 * x1 + 12 * x2 >= 44)
    model.addConstr(9 * x0 + 12 * x2 + 9 * x3 >= 44)
    model.addConstr(9 * x1 + 12 * x2 + 9 * x3 >= 44)
    model.addConstr(9 * x0 + 9 * x1 + 12 * x2 + 9 * x3 >= 44)
    model.addConstr(9 * x0 + 10 * x2 >= 68)
    model.addConstr(11 * x1 + 11 * x3 >= 66)
    model.addConstr(9 * x0 + 11 * x1 >= 76)
    model.addConstr(9 * x0 + 10 * x2 + 11 * x3 >= 76)
    model.addConstr(9 * x0 + 11 * x1 + 10 * x2 + 11 * x3 >= 76)
    model.addConstr(6 * x1 + 2 * x2 >= 13)
    model.addConstr(2 * x2 + 11 * x3 >= 13)
    model.addConstr(11 * x0 + 2 * x2 >= 18)
    model.addConstr(11 * x0 + 6 * x1 + 2 * x2 + 11 * x3 >= 18)
    model.addConstr(-10 * x2 + 3 * x3 >= 0)
    model.addConstr(9 * x1 + 9 * x3 <= 169)
    model.addConstr(9 * x0 + 12 * x2 <= 178)
    model.addConstr(9 * x0 + 9 * x1 <= 228)
    model.addConstr(9 * x1 + 12 * x2 + 9 * x3 <= 91)
    model.addConstr(9 * x0 + 11 * x1 + 11 * x3 <= 86)
    model.addConstr(6 * x1 + 2 * x2 + 11 * x3 <= 26)
    model.addConstr(11 * x0 + 2 * x2 + 11 * x3 <= 62)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Dale: ", x0.varValue)
        print("Hours worked by Paul: ", x1.varValue)
        print("Hours worked by Laura: ", x2.varValue)
        print("Hours worked by John: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```