## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Bill'
- 'x2' corresponds to 'hours worked by Mary'
- 'x3' corresponds to 'hours worked by Dale'
- 'x4' corresponds to 'hours worked by John'
- 'x5' corresponds to 'hours worked by Jean'
- 'x6' corresponds to 'hours worked by Paul'

## Step 2: Define the productivity ratings for each person
The productivity ratings are given as:
- Bill: 20
- Mary: 14
- Dale: 17
- John: 3
- Jean: 20
- Paul: 12

## Step 3: Formulate the objective function
The objective function to minimize is: $9.2x_1 + 4.25x_2 + 6.62x_3 + 4.22x_4 + 8.6x_5 + 9.09x_6$

## 4: List all the constraints
The constraints are:
1. $17x_3 + 3x_4 \geq 41$
2. $20x_1 + 20x_5 \geq 46$
3. $20x_1 + 3x_4 \geq 47$
4. $14x_2 + 12x_6 \geq 47$
5. $14x_2 + 20x_5 \geq 62$
6. $17x_3 + 20x_5 \geq 35$
7. $14x_2 + 17x_3 \geq 48$
8. $20x_5 + 12x_6 \geq 28$
9. $3x_4 + 12x_6 \geq 66$
10. $20x_1 + 14x_2 + 17x_3 + 3x_4 + 20x_5 + 12x_6 \geq 66$
11. $-9x_4 + 10x_5 \geq 0$
12. $-10x_2 + 7x_6 \geq 0$
13. $7x_3 - x_5 \geq 0$
14. $-8x_2 + 3x_4 \geq 0$
15. $17x_3 + 3x_4 \leq 190$
16. $17x_3 + 20x_5 \leq 409$
17. $20x_1 + 20x_5 \leq 130$
18. $3x_4 + 12x_6 \leq 336$
19. $20x_1 + 17x_3 \leq 138$
20. $17x_3 + 12x_6 \leq 182$
21. $20x_5 + 12x_6 \leq 222$
22. $20x_1 + 3x_4 \leq 320$
23. $14x_2 + 17x_3 \leq 276$
24. $14x_2 + 3x_4 \leq 440$
25. $20x_1 + 12x_6 \leq 365$
26. $3x_4 + 20x_5 \leq 171$

## 5: Convert the problem into a Gurobi code
```python
import gurobi

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

# Define the variables
x1 = m.addVar(name='x1', lb=0)  # hours worked by Bill
x2 = m.addVar(name='x2', lb=0)  # hours worked by Mary
x3 = m.addVar(name='x3', lb=0)  # hours worked by Dale
x4 = m.addVar(name='x4', lb=0)  # hours worked by John
x5 = m.addVar(name='x5', lb=0)  # hours worked by Jean
x6 = m.addVar(name='x6', lb=0)  # hours worked by Paul

# Define the objective function
m.setObjective(9.2*x1 + 4.25*x2 + 6.62*x3 + 4.22*x4 + 8.6*x5 + 9.09*x6, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(17*x3 + 3*x4 >= 41)
m.addConstr(20*x1 + 20*x5 >= 46)
m.addConstr(20*x1 + 3*x4 >= 47)
m.addConstr(14*x2 + 12*x6 >= 47)
m.addConstr(14*x2 + 20*x5 >= 62)
m.addConstr(17*x3 + 20*x5 >= 35)
m.addConstr(14*x2 + 17*x3 >= 48)
m.addConstr(20*x5 + 12*x6 >= 28)
m.addConstr(3*x4 + 12*x6 >= 66)
m.addConstr(20*x1 + 14*x2 + 17*x3 + 3*x4 + 20*x5 + 12*x6 >= 66)
m.addConstr(-9*x4 + 10*x5 >= 0)
m.addConstr(-10*x2 + 7*x6 >= 0)
m.addConstr(7*x3 - x5 >= 0)
m.addConstr(-8*x2 + 3*x4 >= 0)
m.addConstr(17*x3 + 3*x4 <= 190)
m.addConstr(17*x3 + 20*x5 <= 409)
m.addConstr(20*x1 + 20*x5 <= 130)
m.addConstr(3*x4 + 12*x6 <= 336)
m.addConstr(20*x1 + 17*x3 <= 138)
m.addConstr(17*x3 + 12*x6 <= 182)
m.addConstr(20*x5 + 12*x6 <= 222)
m.addConstr(20*x1 + 3*x4 <= 320)
m.addConstr(14*x2 + 17*x3 <= 276)
m.addConstr(14*x2 + 3*x4 <= 440)
m.addConstr(20*x1 + 12*x6 <= 365)
m.addConstr(3*x4 + 20*x5 <= 171)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print('Objective: ', m.objval)
    print('x1: ', x1.varValue)
    print('x2: ', x2.varValue)
    print('x3: ', x3.varValue)
    print('x4: ', x4.varValue)
    print('x5: ', x5.varValue)
    print('x6: ', x6.varValue)
else:
    print('No solution found')
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Bill'),
        ('x2', 'hours worked by Mary'),
        ('x3', 'hours worked by Dale'),
        ('x4', 'hours worked by John'),
        ('x5', 'hours worked by Jean'),
        ('x6', 'hours worked by Paul')
    ],
    'objective_function': '9.2*x1 + 4.25*x2 + 6.62*x3 + 4.22*x4 + 8.6*x5 + 9.09*x6',
    'constraints': [
        '17*x3 + 3*x4 >= 41',
        '20*x1 + 20*x5 >= 46',
        '20*x1 + 3*x4 >= 47',
        '14*x2 + 12*x6 >= 47',
        '14*x2 + 20*x5 >= 62',
        '17*x3 + 20*x5 >= 35',
        '14*x2 + 17*x3 >= 48',
        '20*x5 + 12*x6 >= 28',
        '3*x4 + 12*x6 >= 66',
        '20*x1 + 14*x2 + 17*x3 + 3*x4 + 20*x5 + 12*x6 >= 66',
        '-9*x4 + 10*x5 >= 0',
        '-10*x2 + 7*x6 >= 0',
        '7*x3 - x5 >= 0',
        '-8*x2 + 3*x4 >= 0',
        '17*x3 + 3*x4 <= 190',
        '17*x3 + 20*x5 <= 409',
        '20*x1 + 20*x5 <= 130',
        '3*x4 + 12*x6 <= 336',
        '20*x1 + 17*x3 <= 138',
        '17*x3 + 12*x6 <= 182',
        '20*x5 + 12*x6 <= 222',
        '20*x1 + 3*x4 <= 320',
        '14*x2 + 17*x3 <= 276',
        '14*x2 + 3*x4 <= 440',
        '20*x1 + 12*x6 <= 365',
        '3*x4 + 20*x5 <= 171'
    ]
}
```