## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_0$ represents 'hours worked by Peggy'
- $x_1$ represents 'hours worked by Bill'
- $x_2$ represents 'hours worked by Laura'
- $x_3$ represents 'hours worked by Dale'

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $2(x_0 + 1)(x_1 + 1)(x_2) + 8(x_3)$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $14x_0 \geq 14$ (Peggy's productivity rating)
- $5x_0 \geq 5$ (Peggy's organization score)
- $11x_0 \geq 11$ (Peggy's paperwork competence rating)
- $8x_1 \geq 8$ (Bill's productivity rating)
- $5x_1 \geq 5$ (Bill's organization score)
- $13x_1 \geq 13$ (Bill's paperwork competence rating)
- $1x_2 \geq 1$ (Laura's productivity rating)
- $3x_2 \geq 3$ (Laura's organization score)
- $2x_2 \geq 2$ (Laura's paperwork competence rating)
- $6x_3 \geq 6$ (Dale's productivity rating)
- $1x_3 \geq 1$ (Dale's organization score)
- $6x_3 \geq 6$ (Dale's paperwork competence rating)
- $14x_0 + 6x_3 \geq 34$ (Total combined productivity rating from Peggy and Dale)
- $14x_0 + 1x_2 \geq 20$ (Total combined productivity rating from Peggy and Laura)
- $14x_0 + 8x_1 \geq 23$ (Total combined productivity rating from Peggy and Bill)
- $1x_2 + 6x_3 \geq 37$ (Total combined productivity rating from Laura and Dale)
- $14x_0 + 1x_2 + 6x_3 \geq 29$ (Total combined productivity rating from Peggy, Laura, and Dale)
- $8x_1 + 1x_2 + 6x_3 \geq 29$ (Total combined productivity rating from Bill, Laura, and Dale)
- $14x_0 + 1x_2 + 6x_3 \geq 26$ (Total combined productivity rating from Peggy, Laura, and Dale)
- $8x_1 + 1x_2 + 6x_3 \geq 26$ (Total combined productivity rating from Bill, Laura, and Dale)
- $14x_0 + 8x_1 + 1x_2 + 6x_3 \geq 26$ (Total combined productivity rating from all)
- $5x_1 + 1x_3 \geq 30$ (Total combined organization score from Bill and Dale)
- $5x_0 + 1x_3 \geq 20$ (Total combined organization score from Peggy and Dale)
- $5x_0 + 3x_2 \geq 10$ (Total combined organization score from Peggy and Laura)
- $5x_0 + 5x_1 + 1x_3 \geq 18$ (Total combined organization score from Peggy, Bill, and Dale)
- $5x_1 + 3x_2 + 1x_3 \geq 18$ (Total combined organization score from Bill, Laura, and Dale)
- $5x_0 + 5x_1 + 1x_3 \geq 29$ (Total combined organization score from Peggy, Bill, and Dale)
- $5x_1 + 3x_2 + 1x_3 \geq 29$ (Total combined organization score from Bill, Laura, and Dale)
- $5x_0 + 5x_1 + 3x_2 + 1x_3 \geq 29$ (Total combined organization score from all)
- $13x_1 + 6x_3 \geq 10$ (Total combined paperwork competence rating from Bill and Dale)
- $13x_1 + 2x_2 \geq 22$ (Total combined paperwork competence rating from Bill and Laura)
- $2x_2 + 6x_3 \geq 15$ (Total combined paperwork competence rating from Laura and Dale)
- $11x_0 + 6x_3 \geq 25$ (Total combined paperwork competence rating from Peggy and Dale)
- $11x_0 + 2x_2 \geq 29$ (Total combined paperwork competence rating from Peggy and Laura)
- $11x_0 + 13x_1 + 6x_3 \geq 32$ (Total combined paperwork competence rating from Peggy, Bill, and Dale)
- $13x_1 + 2x_2 + 6x_3 \geq 32$ (Total combined paperwork competence rating from Bill, Laura, and Dale)
- $11x_0 + 13x_1 + 6x_3 \geq 32$ (Total combined paperwork competence rating from Peggy, Bill, and Dale)
- $13x_1 + 2x_2 + 6x_3 \geq 32$ (Total combined paperwork competence rating from Bill, Laura, and Dale)
- $11x_0 + 13x_1 + 2x_2 + 6x_3 \geq 32$ (Total combined paperwork competence rating from all)
- $14x_0 + 1x_2 \leq 112$ (Total combined productivity rating from Peggy and Laura)
- $14x_0 + 8x_1 + 1x_2 \leq 133$ (Total combined productivity rating from Peggy, Bill, and Laura)
- $14x_0 + 8x_1 + 6x_3 \leq 148$ (Total combined productivity rating from Peggy, Bill, and Dale)
- $14x_0 + 1x_2 + 6x_3 \leq 172$ (Total combined productivity rating from Peggy, Laura, and Dale)
- $3x_2 + 1x_3 \leq 71$ (Total combined organization score from Laura and Dale)
- $11x_0 + 13x_1 + 6x_3 \leq 34$ (Total combined paperwork competence rating from Peggy, Bill, and Dale)
- $11x_0 + 13x_1 + 2x_2 \leq 109$ (Total combined paperwork competence rating from Peggy, Bill, and Laura)

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

# Create a new Gurobi model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Peggy
x1 = m.addVar(name="x1", lb=0)  # hours worked by Bill
x2 = m.addVar(name="x2", lb=0)  # hours worked by Laura
x3 = m.addVar(name="x3", lb=0)  # hours worked by Dale

# Define the objective function
m.setObjective(2 * (x0 + 1) * (x1 + 1) * x2 + 8 * x3, gurobi.GRB.MINIMIZE)

# Add constraints
# Individual ratings
m.addConstr(14 * x0 >= 14)
m.addConstr(5 * x0 >= 5)
m.addConstr(11 * x0 >= 11)
m.addConstr(8 * x1 >= 8)
m.addConstr(5 * x1 >= 5)
m.addConstr(13 * x1 >= 13)
m.addConstr(x2 >= 1)
m.addConstr(3 * x2 >= 3)
m.addConstr(2 * x2 >= 2)
m.addConstr(6 * x3 >= 6)
m.addConstr(x3 >= 1)
m.addConstr(6 * x3 >= 6)

# Combined productivity ratings
m.addConstr(14 * x0 + 6 * x3 >= 34)
m.addConstr(14 * x0 + x2 >= 20)
m.addConstr(14 * x0 + 8 * x1 >= 23)
m.addConstr(x2 + 6 * x3 >= 37)
m.addConstr(14 * x0 + x2 + 6 * x3 >= 29)
m.addConstr(8 * x1 + x2 + 6 * x3 >= 29)
m.addConstr(14 * x0 + x2 + 6 * x3 >= 26)
m.addConstr(8 * x1 + x2 + 6 * x3 >= 26)
m.addConstr(14 * x0 + 8 * x1 + x2 + 6 * x3 >= 26)

# Combined organization scores
m.addConstr(5 * x1 + x3 >= 30)
m.addConstr(5 * x0 + x3 >= 20)
m.addConstr(5 * x0 + 3 * x2 >= 10)
m.addConstr(5 * x0 + 5 * x1 + x3 >= 18)
m.addConstr(5 * x1 + 3 * x2 + x3 >= 18)
m.addConstr(5 * x0 + 5 * x1 + x3 >= 29)
m.addConstr(5 * x1 + 3 * x2 + x3 >= 29)
m.addConstr(5 * x0 + 5 * x1 + 3 * x2 + x3 >= 29)

# Combined paperwork competence ratings
m.addConstr(13 * x1 + 6 * x3 >= 10)
m.addConstr(13 * x1 + 2 * x2 >= 22)
m.addConstr(2 * x2 + 6 * x3 >= 15)
m.addConstr(11 * x0 + 6 * x3 >= 25)
m.addConstr(11 * x0 + 2 * x2 >= 29)
m.addConstr(11 * x0 + 13 * x1 + 6 * x3 >= 32)
m.addConstr(13 * x1 + 2 * x2 + 6 * x3 >= 32)
m.addConstr(11 * x0 + 13 * x1 + 6 * x3 >= 32)
m.addConstr(13 * x1 + 2 * x2 + 6 * x3 >= 32)
m.addConstr(11 * x0 + 13 * x1 + 2 * x2 + 6 * x3 >= 32)

# Bounds on combined ratings
m.addConstr(14 * x0 + x2 <= 112)
m.addConstr(14 * x0 + 8 * x1 + x2 <= 133)
m.addConstr(14 * x0 + 8 * x1 + 6 * x3 <= 148)
m.addConstr(14 * x0 + x2 + 6 * x3 <= 172)
m.addConstr(3 * x2 + x3 <= 71)
m.addConstr(11 * x0 + 13 * x1 + 6 * x3 <= 34)
m.addConstr(11 * x0 + 13 * x1 + 2 * x2 <= 109)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Hours worked by Peggy: ", x0.varValue)
    print("Hours worked by Bill: ", x1.varValue)
    print("Hours worked by Laura: ", x2.varValue)
    print("Hours worked by Dale: ", x3.varValue)
else:
    print("No optimal solution found")
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'),
        ('x1', 'hours worked by Bill'),
        ('x2', 'hours worked by Laura'),
        ('x3', 'hours worked by Dale')
    ],
    'objective_function': '2 * (x0 + 1) * (x1 + 1) * x2 + 8 * x3',
    'constraints': [
        '14 * x0 >= 14',
        '5 * x0 >= 5',
        '11 * x0 >= 11',
        '8 * x1 >= 8',
        '5 * x1 >= 5',
        '13 * x1 >= 13',
        'x2 >= 1',
        '3 * x2 >= 3',
        '2 * x2 >= 2',
        '6 * x3 >= 6',
        'x3 >= 1',
        '6 * x3 >= 6',
        '14 * x0 + 6 * x3 >= 34',
        '14 * x0 + x2 >= 20',
        '14 * x0 + 8 * x1 >= 23',
        'x2 + 6 * x3 >= 37',
        '14 * x0 + x2 + 6 * x3 >= 29',
        '8 * x1 + x2 + 6 * x3 >= 29',
        '14 * x0 + x2 + 6 * x3 >= 26',
        '8 * x1 + x2 + 6 * x3 >= 26',
        '14 * x0 + 8 * x1 + x2 + 6 * x3 >= 26',
        '5 * x1 + x3 >= 30',
        '5 * x0 + x3 >= 20',
        '5 * x0 + 3 * x2 >= 10',
        '5 * x0 + 5 * x1 + x3 >= 18',
        '5 * x1 + 3 * x2 + x3 >= 18',
        '5 * x0 + 5 * x1 + x3 >= 29',
        '5 * x1 + 3 * x2 + x3 >= 29',
        '5 * x0 + 5 * x1 + 3 * x2 + x3 >= 29',
        '13 * x1 + 6 * x3 >= 10',
        '13 * x1 + 2 * x2 >= 22',
        '2 * x2 + 6 * x3 >= 15',
        '11 * x0 + 6 * x3 >= 25',
        '11 * x0 + 2 * x2 >= 29',
        '11 * x0 + 13 * x1 + 6 * x3 >= 32',
        '13 * x1 + 2 * x2 + 6 * x3 >= 32',
        '11 * x0 + 13 * x1 + 6 * x3 >= 32',
        '13 * x1 + 2 * x2 + 6 * x3 >= 32',
        '11 * x0 + 13 * x1 + 2 * x2 + 6 * x3 >= 32',
        '14 * x0 + x2 <= 112',
        '14 * x0 + 8 * x1 + x2 <= 133',
        '14 * x0 + 8 * x1 + 6 * x3 <= 148',
        '14 * x0 + x2 + 6 * x3 <= 172',
        '3 * x2 + x3 <= 71',
        '11 * x0 + 13 * x1 + 6 * x3 <= 34',
        '11 * x0 + 13 * x1 + 2 * x2 <= 109'
    ]
}
```