## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. Let's define the symbolic variables:
- $x_0$ represents 'hours worked by Laura'
- $x_1$ represents 'hours worked by Hank'
- $x_2$ represents 'hours worked by Bobby'

The objective function to maximize is: $6.81x_0 + 3.82x_1 + 7.83x_2$

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. $2x_0 \leq 197$ (Laura's likelihood to quit index)
2. $16x_0 \leq 146$ (Laura's organization score)
3. $19x_1 \leq 197$ (Hank's likelihood to quit index)
4. $16x_1 \leq 146$ (Hank's organization score)
5. $9x_2 \leq 197$ (Bobby's likelihood to quit index)
6. $15x_2 \leq 146$ (Bobby's organization score)
7. $2x_0 + 19x_1 \geq 60$ (Total combined likelihood to quit index from Laura and Hank)
8. $19x_1 + 9x_2 \leq 192$ (Total combined likelihood to quit index from Hank and Bobby)
9. $2x_0 + 9x_2 \leq 98$ (Total combined likelihood to quit index from Laura and Bobby)
10. $2x_0 + 19x_1 \leq 109$ (Total combined likelihood to quit index from Laura and Hank)
11. $2x_0 + 19x_1 + 9x_2 \leq 109$ (Total combined likelihood to quit index from all)
12. $16x_0 + 15x_2 \leq 139$ (Total combined organization score from Laura and Bobby)
13. $16x_0 + 16x_1 \leq 74$ (Total combined organization score from Laura and Hank)
14. $16x_0 + 16x_1 + 15x_2 \leq 74$ (Total combined organization score from all)

## 3: Convert the problem into a Gurobi model
We will use Gurobi to solve this linear programming problem.

```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Laura", lb=0)  # Laura
x1 = m.addVar(name="hours_worked_by_Hank", lb=0)   # Hank
x2 = m.addVar(name="hours_worked_by_Bobby", lb=0)  # Bobby

# Objective function
m.setObjective(6.81 * x0 + 3.82 * x1 + 7.83 * x2, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(2 * x0 <= 197, name="Laura_likelihood_to_quit")
m.addConstr(16 * x0 <= 146, name="Laura_organization_score")
m.addConstr(19 * x1 <= 197, name="Hank_likelihood_to_quit")
m.addConstr(16 * x1 <= 146, name="Hank_organization_score")
m.addConstr(9 * x2 <= 197, name="Bobby_likelihood_to_quit")
m.addConstr(15 * x2 <= 146, name="Bobby_organization_score")

m.addConstr(2 * x0 + 19 * x1 >= 60, name="Laura_Hank_likelihood_to_quit")
m.addConstr(19 * x1 + 9 * x2 <= 192, name="Hank_Bobby_likelihood_to_quit")
m.addConstr(2 * x0 + 9 * x2 <= 98, name="Laura_Bobby_likelihood_to_quit")
m.addConstr(2 * x0 + 19 * x1 <= 109, name="Laura_Hank_likelihood_to_quit_max")
m.addConstr(2 * x0 + 19 * x1 + 9 * x2 <= 109, name="all_likelihood_to_quit")

m.addConstr(16 * x0 + 15 * x2 <= 139, name="Laura_Bobby_organization_score")
m.addConstr(16 * x0 + 16 * x1 <= 74, name="Laura_Hank_organization_score")
m.addConstr(16 * x0 + 16 * x1 + 15 * x2 <= 74, name="all_organization_score")

# Optimize the model
m.optimize()

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

## 4: Provide the symbolic representation in JSON format
```json
{
    "sym_variables": [
        ["x0", "hours worked by Laura"],
        ["x1", "hours worked by Hank"],
        ["x2", "hours worked by Bobby"]
    ],
    "objective_function": "6.81x0 + 3.82x1 + 7.83x2",
    "constraints": [
        "2x0 <= 197",
        "16x0 <= 146",
        "19x1 <= 197",
        "16x1 <= 146",
        "9x2 <= 197",
        "15x2 <= 146",
        "2x0 + 19x1 >= 60",
        "19x1 + 9x2 <= 192",
        "2x0 + 9x2 <= 98",
        "2x0 + 19x1 <= 109",
        "2x0 + 19x1 + 9x2 <= 109",
        "16x0 + 15x2 <= 139",
        "16x0 + 16x1 <= 74",
        "16x0 + 16x1 + 15x2 <= 74"
    ]
}
```