## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Laura', 'hours worked by Bill', 'hours worked by Hank', and 'hours worked by Paul', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the problem description into a symbolic representation
The objective function to minimize is $7x_0 + 8x_1 + 1x_2 + 3x_3$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $26x_0 \leq 343$
- $9x_0 \leq 184$
- $23x_1 \leq 343$
- $26x_1 \leq 184$
- $12x_2 \leq 343$
- $8x_2 \leq 184$
- $13x_3 \leq 343$
- $19x_3 \leq 184$
- $26x_0 + 12x_2 \geq 70$
- $23x_1 + 12x_2 \geq 76$
- $26x_0 + 23x_1 \geq 70$
- $26x_0 + 23x_1 + 12x_2 + 13x_3 \geq 70$
- $8x_2 + 19x_3 \geq 31$
- $9x_0 + 8x_2 \geq 28$
- $9x_0 + 26x_1 \geq 44$
- $9x_0 + 26x_1 + 19x_3 \geq 28$
- $9x_0 + 26x_1 + 8x_2 + 19x_3 \geq 28$
- $-5x_1 + x_3 \geq 0$
- $-7x_1 + 2x_2 \geq 0$
- $6x_0 - 2x_1 \geq 0$
- $26x_0 + 23x_1 \leq 190$
- $23x_1 + 13x_3 \leq 272$
- $12x_2 + 13x_3 \leq 119$
- $26x_0 + 13x_3 \leq 108$
- $9x_0 + 26x_1 + 19x_3 \leq 168$

## 4: Create the symbolic representation dictionary
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'), 
        ('x1', 'hours worked by Bill'), 
        ('x2', 'hours worked by Hank'), 
        ('x3', 'hours worked by Paul')
    ], 
    'objective_function': '7*x0 + 8*x1 + 1*x2 + 3*x3', 
    'constraints': [
        '26*x0 <= 343', '9*x0 <= 184', '23*x1 <= 343', '26*x1 <= 184', 
        '12*x2 <= 343', '8*x2 <= 184', '13*x3 <= 343', '19*x3 <= 184', 
        '26*x0 + 12*x2 >= 70', '23*x1 + 12*x2 >= 76', '26*x0 + 23*x1 >= 70', 
        '26*x0 + 23*x1 + 12*x2 + 13*x3 >= 70', '8*x2 + 19*x3 >= 31', 
        '9*x0 + 8*x2 >= 28', '9*x0 + 26*x1 >= 44', '9*x0 + 26*x1 + 19*x3 >= 28', 
        '9*x0 + 26*x1 + 8*x2 + 19*x3 >= 28', '-5*x1 + x3 >= 0', 
        '-7*x1 + 2*x2 >= 0', '6*x0 - 2*x1 >= 0', '26*x0 + 23*x1 <= 190', 
        '23*x1 + 13*x3 <= 272', '12*x2 + 13*x3 <= 119', '26*x0 + 13*x3 <= 108', 
        '9*x0 + 26*x1 + 19*x3 <= 168'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

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

    # Define the objective function
    model.setObjective(7 * x0 + 8 * x1 + 1 * x2 + 3 * x3, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(26 * x0 <= 343)
    model.addConstr(9 * x0 <= 184)
    model.addConstr(23 * x1 <= 343)
    model.addConstr(26 * x1 <= 184)
    model.addConstr(12 * x2 <= 343)
    model.addConstr(8 * x2 <= 184)
    model.addConstr(13 * x3 <= 343)
    model.addConstr(19 * x3 <= 184)
    model.addConstr(26 * x0 + 12 * x2 >= 70)
    model.addConstr(23 * x1 + 12 * x2 >= 76)
    model.addConstr(26 * x0 + 23 * x1 >= 70)
    model.addConstr(26 * x0 + 23 * x1 + 12 * x2 + 13 * x3 >= 70)
    model.addConstr(8 * x2 + 19 * x3 >= 31)
    model.addConstr(9 * x0 + 8 * x2 >= 28)
    model.addConstr(9 * x0 + 26 * x1 >= 44)
    model.addConstr(9 * x0 + 26 * x1 + 19 * x3 >= 28)
    model.addConstr(9 * x0 + 26 * x1 + 8 * x2 + 19 * x3 >= 28)
    model.addConstr(-5 * x1 + x3 >= 0)
    model.addConstr(-7 * x1 + 2 * x2 >= 0)
    model.addConstr(6 * x0 - 2 * x1 >= 0)
    model.addConstr(26 * x0 + 23 * x1 <= 190)
    model.addConstr(23 * x1 + 13 * x3 <= 272)
    model.addConstr(12 * x2 + 13 * x3 <= 119)
    model.addConstr(26 * x0 + 13 * x3 <= 108)
    model.addConstr(9 * x0 + 26 * x1 + 19 * x3 <= 168)

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```