## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x0' corresponds to 'hours worked by Paul'
- 'x1' corresponds to 'hours worked by Bobby'
- 'x2' corresponds to 'hours worked by Peggy'
- 'x3' corresponds to 'hours worked by Jean'
- 'x4' corresponds to 'hours worked by Dale'

## Step 2: Define the symbolic representation of the problem
The symbolic representation of the variables is:
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Paul'), 
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Peggy'), 
        ('x3', 'hours worked by Jean'), 
        ('x4', 'hours worked by Dale')
    ],
}
```

## Step 3: Define the objective function in symbolic notation
The objective function to maximize is: 
\[ 8 \times \text{hours worked by Paul} + 6 \times \text{hours worked by Bobby} + 2 \times \text{hours worked by Peggy} + 5 \times \text{hours worked by Jean} + 3 \times \text{hours worked by Dale} \]
In symbolic notation, this is:
\[ 8x0 + 6x1 + 2x2 + 5x3 + 3x4 \]

## 4: List the constraints in symbolic notation
The constraints are:
- \( 11x0 \leq 11 \) (Paul's computer competence rating is 11)
- \( 12x0 \leq 12 \) (Paul's paperwork competence rating is 12)
- \( 6x0 \leq 6 \) (Paul's work quality rating is 6)
- \( 14x1 \leq 14 \) (Bobby's computer competence rating is 14)
- \( 14x1 \leq 14 \) (Bobby's paperwork competence rating is 14)
- \( 6x1 \leq 6 \) (Bobby's work quality rating is 6)
- \( 10x2 \leq 10 \) (Peggy's computer competence rating is 10)
- \( 11x2 \leq 11 \) (Peggy's paperwork competence rating is 11)
- \( 3x2 \leq 3 \) (Peggy's work quality rating is 3)
- \( 10x3 \leq 10 \) (Jean's computer competence rating is 10)
- \( 10x3 \leq 10 \) (Jean's paperwork competence rating is 10)
- \( 7x3 \leq 7 \) (Jean's work quality rating is 7)
- \( 5x4 \leq 5 \) (Dale's computer competence rating is 5)
- \( 1x4 \leq 1 \) (Dale's paperwork competence rating is 1)
- \( 8x4 \leq 8 \) (Dale's work quality rating is 8)
- \( 11x0 + 14x1 + 10x2 \geq 32 \) (total combined computer competence rating from hours worked by Paul, Bobby, and Peggy)
- \( 11x0 + 10x2 \leq 123 \) (total combined computer competence rating from hours worked by Paul and Peggy)
- \( 14x1 + 5x4 \leq 87 \) (total combined computer competence rating from hours worked by Bobby and Dale)
- \( 14x1 + 10x2 \leq 151 \) (total combined computer competence rating from hours worked by Bobby and Peggy)
- \( 10x2 + 5x4 \leq 143 \) (total combined computer competence rating from hours worked by Peggy and Dale)
- \( 11x0 + 5x4 \leq 125 \) (total combined computer competence rating from hours worked by Paul and Dale)
- \( 11x0 + 10x3 + 5x4 \leq 82 \) (total combined computer competence rating from hours worked by Paul, Jean, and Dale)
- \( 11x0 + 14x1 + 10x2 + 10x3 + 5x4 \leq 82 \) (total combined computer competence rating from all)
- \( 14x1 + 10x3 \leq 221 \) (total combined paperwork competence rating from hours worked by Bobby and Jean)
- \( 10x3 + 1x4 \leq 326 \) (total combined paperwork competence rating from hours worked by Jean and Dale)
- \( 12x0 + 14x1 + 11x2 + 10x3 + 1x4 \leq 326 \) (total combined paperwork competence rating from all)
- \( 6x0 + 7x3 \leq 77 \) (total combined work quality rating from hours worked by Paul and Jean)
- \( 3x2 + 7x3 \leq 62 \) (total combined work quality rating from hours worked by Peggy and Jean)
- \( 7x3 + 8x4 \leq 75 \) (total combined work quality rating from hours worked by Jean and Dale)
- \( 6x0 + 3x2 \leq 84 \) (total combined work quality rating from hours worked by Paul and Peggy)
- \( 6x1 + 3x2 \leq 37 \) (total combined work quality rating from hours worked by Bobby and Peggy)
- \( 6x0 + 8x4 \leq 60 \) (total combined work quality rating from hours worked by Paul and Dale)
- \( 3x2 + 7x3 + 8x4 \leq 40 \) (total combined work quality rating from hours worked by Peggy, Jean, and Dale)
- \( 6x1 + 3x2 + 8x4 \leq 77 \) (total combined work quality rating from hours worked by Bobby, Peggy, and Dale)
- \( 6x1 + 3x2 + 7x3 \leq 76 \) (total combined work quality rating from hours worked by Bobby, Peggy, and Jean)
- \( 6x0 + 3x2 + 8x4 \leq 52 \) (total combined work quality rating from hours worked by Paul, Peggy, and Dale)
- \( 6x0 + 3x2 + 7x3 \leq 33 \) (total combined work quality rating from hours worked by Paul, Peggy, and Jean)
- \( 6x1 + 7x3 + 8x4 \leq 17 \) (total combined work quality rating from hours worked by Bobby, Jean, and Dale)
- \( 6x0 + 6x1 + 3x2 + 7x3 + 8x4 \leq 17 \) (total combined work quality rating from all)

## 5: Write down the full problem description in JSON
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Paul'), 
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Peggy'), 
        ('x3', 'hours worked by Jean'), 
        ('x4', 'hours worked by Dale')
    ],
    'objective_function': '8*x0 + 6*x1 + 2*x2 + 5*x3 + 3*x4',
    'constraints': [
        '11*x0 <= 11',
        '12*x0 <= 12',
        '6*x0 <= 6',
        '14*x1 <= 14',
        '14*x1 <= 14',
        '6*x1 <= 6',
        '10*x2 <= 10',
        '11*x2 <= 11',
        '3*x2 <= 3',
        '10*x3 <= 10',
        '10*x3 <= 10',
        '7*x3 <= 7',
        '5*x4 <= 5',
        '1*x4 <= 1',
        '8*x4 <= 8',
        '11*x0 + 14*x1 + 10*x2 >= 32',
        '11*x0 + 10*x2 <= 123',
        '14*x1 + 5*x4 <= 87',
        '14*x1 + 10*x2 <= 151',
        '10*x2 + 5*x4 <= 143',
        '11*x0 + 5*x4 <= 125',
        '11*x0 + 10*x3 + 5*x4 <= 82',
        '11*x0 + 14*x1 + 10*x2 + 10*x3 + 5*x4 <= 82',
        '14*x1 + 10*x3 <= 221',
        '10*x3 + 1*x4 <= 326',
        '12*x0 + 14*x1 + 11*x2 + 10*x3 + 1*x4 <= 326',
        '6*x0 + 7*x3 <= 77',
        '3*x2 + 7*x3 <= 62',
        '7*x3 + 8*x4 <= 75',
        '6*x0 + 3*x2 <= 84',
        '6*x1 + 3*x2 <= 37',
        '6*x0 + 8*x4 <= 60',
        '3*x2 + 7*x3 + 8*x4 <= 40',
        '6*x1 + 3*x2 + 8*x4 <= 77',
        '6*x1 + 3*x2 + 7*x3 <= 76',
        '6*x0 + 3*x2 + 8*x4 <= 52',
        '6*x0 + 3*x2 + 7*x3 <= 33',
        '6*x1 + 7*x3 + 8*x4 <= 17',
        '6*x0 + 6*x1 + 3*x2 + 7*x3 + 8*x4 <= 17'
    ]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Paul
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Bobby
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Peggy
    x3 = model.addVar(name="x3", lb=0, integrality=1)  # hours worked by Jean
    x4 = model.addVar(name="x4", lb=0)  # hours worked by Dale

    # Objective function
    model.setObjective(8 * x0 + 6 * x1 + 2 * x2 + 5 * x3 + 3 * x4, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(11 * x0 <= 11)
    model.addConstr(12 * x0 <= 12)
    model.addConstr(6 * x0 <= 6)
    model.addConstr(14 * x1 <= 14)
    model.addConstr(14 * x1 <= 14)
    model.addConstr(6 * x1 <= 6)
    model.addConstr(10 * x2 <= 10)
    model.addConstr(11 * x2 <= 11)
    model.addConstr(3 * x2 <= 3)
    model.addConstr(10 * x3 <= 10)
    model.addConstr(10 * x3 <= 10)
    model.addConstr(7 * x3 <= 7)
    model.addConstr(5 * x4 <= 5)
    model.addConstr(1 * x4 <= 1)
    model.addConstr(8 * x4 <= 8)
    model.addConstr(11 * x0 + 14 * x1 + 10 * x2 >= 32)
    model.addConstr(11 * x0 + 10 * x2 <= 123)
    model.addConstr(14 * x1 + 5 * x4 <= 87)
    model.addConstr(14 * x1 + 10 * x2 <= 151)
    model.addConstr(10 * x2 + 5 * x4 <= 143)
    model.addConstr(11 * x0 + 5 * x4 <= 125)
    model.addConstr(11 * x0 + 10 * x3 + 5 * x4 <= 82)
    model.addConstr(11 * x0 + 14 * x1 + 10 * x2 + 10 * x3 + 5 * x4 <= 82)
    model.addConstr(14 * x1 + 10 * x3 <= 221)
    model.addConstr(10 * x3 + 1 * x4 <= 326)
    model.addConstr(12 * x0 + 14 * x1 + 11 * x2 + 10 * x3 + 1 * x4 <= 326)
    model.addConstr(6 * x0 + 7 * x3 <= 77)
    model.addConstr(3 * x2 + 7 * x3 <= 62)
    model.addConstr(7 * x3 + 8 * x4 <= 75)
    model.addConstr(6 * x0 + 3 * x2 <= 84)
    model.addConstr(6 * x1 + 3 * x2 <= 37)
    model.addConstr(6 * x0 + 8 * x4 <= 60)
    model.addConstr(3 * x2 + 7 * x3 + 8 * x4 <= 40)
    model.addConstr(6 * x1 + 3 * x2 + 8 * x4 <= 77)
    model.addConstr(6 * x1 + 3 * x2 + 7 * x3 <= 76)
    model.addConstr(6 * x0 + 3 * x2 + 8 * x4 <= 52)
    model.addConstr(6 * x0 + 3 * x2 + 7 * x3 <= 33)
    model.addConstr(6 * x1 + 7 * x3 + 8 * x4 <= 17)
    model.addConstr(6 * x0 + 6 * x1 + 3 * x2 + 7 * x3 + 8 * x4 <= 17)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Paul: ", x0.varValue)
        print("Hours worked by Bobby: ", x1.varValue)
        print("Hours worked by Peggy: ", x2.varValue)
        print("Hours worked by Jean: ", x3.varValue)
        print("Hours worked by Dale: ", x4.varValue)
    else:
        print("The problem is infeasible")

optimize_problem()
```