## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy', 'hours worked by Paul', and 'hours worked by Bobby'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $7.9x_0 + 9.83x_1 + 7.21x_2$.

## Step 3: List the constraints
The constraints are:
- $4x_0 \leq 208$ (likelihood to quit index for Peggy, but this seems to be an attribute and not a constraint on $x_0$ directly. Instead, we consider it as a given attribute: $r0_{x0} = 4$)
- $18x_0 \leq 129$ (productivity rating for Peggy, similarly, $r1_{x0} = 18$)
- $20x_0 \leq 221$ (computer competence rating for Peggy, $r2_{x0} = 20$)
- $6x_1 \leq 208$ (likelihood to quit index for Paul, $r0_{x1} = 6$)
- $13x_1 \leq 129$ (productivity rating for Paul, $r1_{x1} = 13$)
- $13x_1 \leq 221$ (computer competence rating for Paul, $r2_{x1} = 13$)
- $1x_2 \leq 208$ (likelihood to quit index for Bobby, $r0_{x2} = 1$)
- $21x_2 \leq 129$ (productivity rating for Bobby, $r1_{x2} = 21$)
- $7x_2 \leq 221$ (computer competence rating for Bobby, $r2_{x2} = 7$)
- $18x_0 + 13x_1 \geq 29$ (combined productivity rating for Peggy and Paul)
- $20x_0 + 13x_1 \geq 59$ (combined computer competence rating for Peggy and Paul)
- $4x_0 + 1x_2 \leq 177$ (combined likelihood to quit index for Peggy and Bobby)
- $4x_0 + 6x_1 \leq 187$ (combined likelihood to quit index for Peggy and Paul)
- $6x_1 + 1x_2 \leq 167$ (combined likelihood to quit index for Paul and Bobby)
- $4x_0 + 6x_1 + 1x_2 \leq 191$ (combined likelihood to quit index for all)
- $18x_0 + 13x_1 \leq 107$ (combined productivity rating for Peggy and Paul)
- $13x_1 + 21x_2 \leq 62$ (combined productivity rating for Paul and Bobby)
- $18x_0 + 13x_1 + 21x_2 \leq 62$ (combined productivity rating for all)
- $13x_1 + 7x_2 \leq 150$ (combined computer competence rating for Paul and Bobby)
- $20x_0 + 13x_1 \leq 208$ (combined computer competence rating for Peggy and Paul)
- $20x_0 + 7x_2 \leq 206$ (combined computer competence rating for Peggy and Bobby)
- $20x_0 + 13x_1 + 7x_2 \leq 206$ (combined computer competence rating for all)

## 4: Correctly formulate the problem in mathematical terms
Given:
- $r0_{x0} = 4$, $r1_{x0} = 18$, $r2_{x0} = 20$
- $r0_{x1} = 6$, $r1_{x1} = 13$, $r2_{x1} = 13$
- $r0_{x2} = 1$, $r1_{x2} = 21$, $r2_{x2} = 7$

Objective: Maximize $7.9x_0 + 9.83x_1 + 7.21x_2$

Constraints:
1. $18x_0 + 13x_1 \geq 29$
2. $20x_0 + 13x_1 \geq 59$
3. $4x_0 + x_2 \leq 177$
4. $4x_0 + 6x_1 \leq 187$
5. $6x_1 + x_2 \leq 167$
6. $4x_0 + 6x_1 + x_2 \leq 191$
7. $18x_0 + 13x_1 \leq 107$
8. $13x_1 + 21x_2 \leq 62$
9. $18x_0 + 13x_1 + 21x_2 \leq 62$
10. $13x_1 + 7x_2 \leq 150$
11. $20x_0 + 13x_1 \leq 208$
12. $20x_0 + 7x_2 \leq 206$
13. $20x_0 + 13x_1 + 7x_2 \leq 206$

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

# Create a new 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 Paul
x2 = m.addVar(name="x2", lb=0)  # hours worked by Bobby

# Define the objective function
m.setObjective(7.9*x0 + 9.83*x1 + 7.21*x2, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(18*x0 + 13*x1 >= 29)
m.addConstr(20*x0 + 13*x1 >= 59)
m.addConstr(4*x0 + x2 <= 177)
m.addConstr(4*x0 + 6*x1 <= 187)
m.addConstr(6*x1 + x2 <= 167)
m.addConstr(4*x0 + 6*x1 + x2 <= 191)
m.addConstr(18*x0 + 13*x1 <= 107)
m.addConstr(13*x1 + 21*x2 <= 62)
m.addConstr(18*x0 + 13*x1 + 21*x2 <= 62)
m.addConstr(13*x1 + 7*x2 <= 150)
m.addConstr(20*x0 + 13*x1 <= 208)
m.addConstr(20*x0 + 7*x2 <= 206)
m.addConstr(20*x0 + 13*x1 + 7*x2 <= 206)

# Optimize 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 Paul: ", x1.varValue)
    print("Hours worked by Bobby: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Peggy'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Bobby')],
    'objective_function': '7.9*x0 + 9.83*x1 + 7.21*x2',
    'constraints': [
        '18*x0 + 13*x1 >= 29',
        '20*x0 + 13*x1 >= 59',
        '4*x0 + x2 <= 177',
        '4*x0 + 6*x1 <= 187',
        '6*x1 + x2 <= 167',
        '4*x0 + 6*x1 + x2 <= 191',
        '18*x0 + 13*x1 <= 107',
        '13*x1 + 21*x2 <= 62',
        '18*x0 + 13*x1 + 21*x2 <= 62',
        '13*x1 + 7*x2 <= 150',
        '20*x0 + 13*x1 <= 208',
        '20*x0 + 7*x2 <= 206',
        '20*x0 + 13*x1 + 7*x2 <= 206'
    ]
}
```