## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Bobby', and 'hours worked by Peggy', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Map the symbolic variables to their natural language counterparts
The symbolic variables and their natural language counterparts are:
- $x_0$ : hours worked by Hank
- $x_1$ : hours worked by Bobby
- $x_2$ : hours worked by Peggy

## 3: Define the objective function in symbolic notation
The objective function to minimize is $9.06x_0 + 5.38x_1 + 9.94x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $6.31x_0 \geq 0$ (not a constraint as it's a variable's attribute, not a constraint)
- $14.73x_1 \geq 0$ (not a constraint as it's a variable's attribute, not a constraint)
- $14.29x_2 \geq 0$ (not a constraint as it's a variable's attribute, not a constraint)
- $6.31x_0 + 14.73x_1 \geq 56$
- $6.31x_0 + 14.73x_1 + 14.29x_2 \geq 56$
- $17.26x_0 + 19.79x_1 \geq 27$
- $17.26x_0 + 19.79x_1 + 9.37x_2 \geq 27$
- $5.05x_1 + 19.88x_2 \geq 30$
- $15.74x_0 + 19.88x_2 \geq 44$
- $15.74x_0 + 5.05x_1 + 19.88x_2 \geq 44$
- $9.25x_0 + 2.08x_2 \geq 22$
- $16.92x_1 + 2.08x_2 \geq 42$
- $9.25x_0 + 16.92x_1 + 2.08x_2 \geq 42$
- $2.6x_0 + 17.17x_2 \geq 38$
- $2.6x_0 + 0.19x_1 + 17.17x_2 \geq 49$
- $2.6x_0 + 0.19x_1 + 17.17x_2 \geq 49$
- $-6x_0 + 6x_1 \geq 0$
- $9.25x_0 + 2.08x_2 \leq 77$
- $2.6x_0 + 0.19x_1 + 17.17x_2 \leq 159$

## 5: Determine the variable types
- $x_0$ : Continuous
- $x_1$ : Integer
- $x_2$ : Continuous

## 6: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(lb=0, name="hours_worked_by_Hank", vtype=gp.GRB.CONTINUOUS)
x1 = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="hours_worked_by_Bobby")
x2 = m.addVar(lb=0, name="hours_worked_by_Peggy", vtype=gp.GRB.CONTINUOUS)

# Define the objective function
m.setObjective(9.06 * x0 + 5.38 * x1 + 9.94 * x2, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(6.31 * x0 + 14.73 * x1 >= 56, name="work_quality_rating_Hank_Bobby")
m.addConstr(6.31 * x0 + 14.73 * x1 + 14.29 * x2 >= 56, name="work_quality_rating_all")
m.addConstr(17.26 * x0 + 19.79 * x1 >= 27, name="likelihood_to_quit_index_Hank_Bobby")
m.addConstr(17.26 * x0 + 19.79 * x1 + 9.37 * x2 >= 27, name="likelihood_to_quit_index_all")
m.addConstr(5.05 * x1 + 19.88 * x2 >= 30, name="dollar_cost_per_hour_Bobby_Peggy")
m.addConstr(15.74 * x0 + 19.88 * x2 >= 44, name="dollar_cost_per_hour_Hank_Peggy")
m.addConstr(15.74 * x0 + 5.05 * x1 + 19.88 * x2 >= 44, name="dollar_cost_per_hour_all")
m.addConstr(9.25 * x0 + 2.08 * x2 >= 22, name="organization_score_Hank_Peggy")
m.addConstr(16.92 * x1 + 2.08 * x2 >= 42, name="organization_score_Bobby_Peggy")
m.addConstr(9.25 * x0 + 16.92 * x1 + 2.08 * x2 >= 42, name="organization_score_all")
m.addConstr(2.6 * x0 + 17.17 * x2 >= 38, name="productivity_rating_Hank_Peggy")
m.addConstr(2.6 * x0 + 0.19 * x1 + 17.17 * x2 >= 49, name="productivity_rating_all")
m.addConstr(-6 * x0 + 6 * x1 >= 0, name="Hank_Bobby_work_balance")
m.addConstr(9.25 * x0 + 2.08 * x2 <= 77, name="organization_score_limit")
m.addConstr(2.6 * x0 + 0.19 * x1 + 17.17 * x2 <= 159, name="productivity_rating_limit")

# Solve the model
m.optimize()

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

```json
{
    'sym_variables': [
        ['x0', 'hours worked by Hank'], 
        ['x1', 'hours worked by Bobby'], 
        ['x2', 'hours worked by Peggy']
    ], 
    'objective_function': '9.06x0 + 5.38x1 + 9.94x2', 
    'constraints': [
        '6.31x0 + 14.73x1 >= 56', 
        '6.31x0 + 14.73x1 + 14.29x2 >= 56', 
        '17.26x0 + 19.79x1 >= 27', 
        '17.26x0 + 19.79x1 + 9.37x2 >= 27', 
        '5.05x1 + 19.88x2 >= 30', 
        '15.74x0 + 19.88x2 >= 44', 
        '15.74x0 + 5.05x1 + 19.88x2 >= 44', 
        '9.25x0 + 2.08x2 >= 22', 
        '16.92x1 + 2.08x2 >= 42', 
        '9.25x0 + 16.92x1 + 2.08x2 >= 42', 
        '2.6x0 + 17.17x2 >= 38', 
        '2.6x0 + 0.19x1 + 17.17x2 >= 49', 
        '-6x0 + 6x1 >= 0', 
        '9.25x0 + 2.08x2 <= 77', 
        '2.6x0 + 0.19x1 + 17.17x2 <= 159'
    ]
}
```